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The Continuum Hypothesis

The continuum hypothesis (CH) is one of the most central open problems in set theory, one that is important for both mathematical and philosophical reasons.

The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. In 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic numbers. More surprisingly, he showed that there is no one-to-one correspondence between the natural numbers and the real numbers. Taking the existence of a one-to-one correspondence as a criterion for when two sets have the same size (something he certainly did by 1878), this result shows that there is more than one level of infinity and thus gave birth to the higher infinite in mathematics. Cantor immediately tried to determine whether there were any infinite sets of real numbers that were of intermediate size, that is, whether there was an infinite set of real numbers that could not be put into one-to-one correspondence with the natural numbers and could not be put into one-to-one correspondence with the real numbers. The continuum hypothesis (under one formulation) is simply the statement that there is no such set of real numbers. It was through his attempt to prove this hypothesis that led Cantor do develop set theory into a sophisticated branch of mathematics. [ 1 ]

Despite his efforts Cantor could not resolve CH. The problem persisted and was considered so important by Hilbert that he placed it first on his famous list of open problems to be faced by the 20 th century. Hilbert also struggled to resolve CH, again without success. Ultimately, this lack of progress was explained by the combined results of Gödel and Cohen, which together showed that CH cannot be resolved on the basis of the axioms that mathematicians were employing; in modern terms, CH is independent of Zermelo-Fraenkel set theory extended with the Axiom of Choice (ZFC).

This independence result was quickly followed by many others. The independence techniques were so powerful that set theorists soon found themselves preoccupied with the meta-theoretic enterprise of proving that certain fundamental statements could not be proved or refuted within ZFC. The question then arose as to whether there were ways to settle the independent statements. The community of mathematicians and philosophers of mathematics was largely divided on this question. The pluralists (like Cohen) maintained that the independence results effectively settled the question by showing that it had no answer . On this view, one could adopt a system in which, say CH was an axiom and one could adopt a system in which ¬CH was an axiom and that was the end of the matter—there was no question as to which of two incompatible extensions was the “correct” one. The non-pluralists (like Gödel) held that the independence results merely indicated the paucity of our means for circumscribing mathematical truth. On this view, what was needed were new axioms, axioms that are both justified and sufficient for the task. Gödel actually went further in proposing candidates for new axioms—large cardinal axioms—and he conjectured that they would settle CH.

Gödel's program for large cardinal axioms proved to be remarkably successful. Over the course of the next 30 years it was shown that large cardinal axioms settle many of the questions that were shown to be independent during the era of independence. However, CH was left untouched. The situation turned out to be rather ironic since in the end it was shown (in a sense that can be made precise) that although the standard large cardinal axioms effectively settle all question of complexity strictly below that of CH, they cannot (by results of Levy and Solovay and others) settle CH itself. Thus, in choosing CH as a test case for his program, Gödel put his finger precisely on the point where it fails. It is for this reason that CH continues to play a central role in the search for new axioms.

In this entry we shall give an overview of the major approaches to settling CH and we shall discuss some of the major foundational frameworks which maintain that CH does not have an answer. The subject is a large one and we have had to sacrifice full comprehensiveness in two dimensions. First, we have not been able to discuss the major philosophical issues that are lying in the background. For this the reader is directed to the entry “ Large Cardinals and Determinacy ”, which contains a general discussion of the independence results, the nature of axioms, the nature of justification, and the successes of large cardinal axioms in the realm “below CH”. Second, we have not been able to discuss every approach to CH that is in the literature. Instead we have restricted ourselves to those approaches that appear most promising from a philosophical point of view and where the mathematics has been developed to a sufficiently advanced state. In the approaches we shall discuss—forcing axioms, inner model theory, quasi-large cardinals—the mathematics has been pressed to a very advanced stage over the course of 40 years. And this has made our task somewhat difficult. We have tried to keep the discussion as accessible as possible and we have placed the more technical items in the endnotes. But the reader should bear in mind that we are presenting a bird's eye view and that for a higher resolution at any point the reader should dip into the suggested readings that appear at the end of each section. [ 2 ]

There are really two kinds of approaches to new axioms—the local approach and the global approach. On the local approach one seeks axioms that answer questions concerning a specifiable fragment of the universe, such as V ω+1 or V ω+2 , where CH lies. On the global approach one seeks axioms that attempt to illuminate the entire structure of the universe of sets. The global approach is clearly much more challenging. In this entry we shall start with the local approach and toward the end we shall briefly touch upon the global approach.

Here is an overview of the entry: Section 1 surveys the independence results in cardinal arithmetic, covering both the case of regular cardinals (where CH lies) and singular cardinals. Section 2 considers approaches to CH where one successively verifies a hierarchy of approximations to CH, each of which is an “effective” version of CH. This approach led to the remarkable discovery of Woodin that it is possible (in the presence of large cardinals) to have an effective failure of CH, thereby showing, that the effective failure of CH is as intractable (with respect to large cardinal axioms) as CH itself. Section 3 continues with the developments that stemmed from this discovery. The centerpiece of the discussion is the discovery of a “canonical” model in which CH fails. This formed the basis of a network of results that was collectively presented by Woodin as a case for the failure of CH. To present this case in the most streamlined form we introduce the strong logic Ω-logic. Section 4 takes up the competing foundational view that there is no solution to CH. This view is sharpened in terms of the generic multiverse conception of truth and that view is then scrutinized. Section 5 continues the assessment of the case for ¬CH by investigating a parallel case for CH. In the remaining two sections we turn to the global approach to new axioms and here we shall be much briefer. Section 6 discusses the approach through inner model theory. Section 7 discusses the approach through quasi-large cardinal axioms.

1.1 Regular Cardinals

1.2 singular cardinals, 2.1 three versions, 2.2 the foreman-magidor program, 3.1 ℙ max, 3.2 ω-logic, 3.3 the case, 4.1 broad multiverse views, 4.2 the generic multiverse, 4.3 the ω conjecture and the generic multiverse, 4.4 is there a way out, 5.1 the case for ¬ch, 5.2 the parallel case for ch, 5.3 assessment.

• 6 The Ultimate Inner Model
• 7 The Structure Theory of L ( V λ+1 )

Other Internet Resources

Related entries, 1. independence in cardinal arithmetic.

In this section we shall discuss the independence results in cardinal arithmetic. First, we shall treat of the case of regular cardinals, where CH lies and where very little is determined in the context of ZFC. Second, for the sake of comprehensiveness, we shall discuss the case of singular cardinals, where much more can be established in the context of ZFC.

The addition and multiplication of infinite cardinal numbers is trivial: For infinite cardinals κ and λ,

κ + λ = κ ⋅ λ = max{κ,λ}.

The situation becomes interesting when one turns to exponentiation and the attempt to compute κ λ for infinite cardinals.

During the dawn of set theory Cantor showed that for every cardinal κ,

2 κ > κ.

There is no mystery about the size of 2 n for finite n . The first natural question then is where 2 ℵ 0 is located in the aleph-hierarchy: Is it ℵ 1 , ℵ 2 , …, ℵ 17 or something much larger?

The cardinal 2 ℵ 0 is important since it is the size of the continuum (the set of real numbers). Cantor's famous continuum hypothesis (CH) is the statement that 2 ℵ 0 = ℵ 1 . This is a special case of the generalized continuum hypothesis (GCH) which asserts that for all α, 2 ℵ α = ℵ α+1 . One virtue of GCH is that it gives a complete solution to the problem of computing κ λ for infinite cardinals: Assuming GCH, if κ ≤ λ then κ λ = λ + ; if cf(κ) ≤ λ ≤ κ then κ λ = κ + ; and if λ < cf(κ) then κ λ = κ.

Very little progress was made on CH and GCH. In fact, in the early era of set theory the only other piece of progress beyond Cantor's result that 2 κ > κ (and the trivial result that if κ ≤ λ then 2 κ ≤ 2 λ ) was König's result that cf(2 κ ) > κ. The explanation for the lack of progress was provided by the independence results in set theory:

To prove this Gödel invented the method of inner models —he showed that CH and GCH held in the minimal inner model L of ZFC. Cohen then complemented this result:

He did this by inventing the method of outer models and showing that CH failed in a generic extension V B of V . The combined results of Gödel and Cohen thus demonstrate that assuming the consistency of ZFC, it is in principle impossible to settle either CH or GCH in ZFC.

In the Fall of 1963 Easton completed the picture by showing that for infinite regular cardinals κ the only constraints on the function κ ↦ 2 κ that are provable in ZFC are the trivial constraint and the results of Cantor and König:

• if κ ≤ λ then F (κ) ≤ F (λ) ,
• F (κ) > κ , and
• cf( F (κ)) > κ .

Thus, set theorists had pushed the cardinal arithmetic of regular cardinals as far as it could be pushed within the confines of ZFC.

The case of cardinal arithmetic on singular cardinals is much more subtle. For the sake of completeness we pause to briefly discuss this before proceeding with the continuum hypothesis.

It was generally believed that, as in the case for regular cardinals, the behaviour of the function κ ↦ 2 κ would be relatively unconstrained within the setting of ZFC. But then Silver proved the following remarkable result: [ 3 ]

It turns out that (by a deep result of Magidor, published in 1977) GCH can first fail at ℵ ω (assuming the consistency of a supercompact cardinal). Silver's theorem shows that it cannot first fail at ℵ ω 1 and this is provable in ZFC.

This raises the question of whether one can “control” the size of 2 ℵ δ with a weaker assumption than that ℵ δ is a singular cardinal of uncountable cofinality such that GCH holds below ℵ δ . The natural hypothesis to consider is that ℵ δ is a singular cardinal of uncountable cofinality which is a strong limit cardinal , that is, that for all α < ℵ δ , 2 α < ℵ δ . In 1975 Galvin and Hajnal proved (among other things) that under this weaker assumption there is indeed a bound:

2 ℵ δ < ℵ (|δ| cf(δ) ) + .

It is possible that there is a jump—in fact, Woodin showed (again assuming large cardinals) that it is possible that for all κ, 2 κ = κ ++ . What the above theorem shows is that in ZFC there is a provable bound on how big the jump can be.

The next question is whether a similar situation prevails with singular cardinals of countable cofinality. In 1978 Shelah showed that this is indeed the case. To fix ideas let us concentrate on ℵ ω .

2 ℵ ω < ℵ (2 ℵ 0 ) + .

One drawback of this result is that the bound is sensitive to the actual size of 2 ℵ 0 , which can be anything below ℵ ω . Remarkably Shelah was later able to remedy this with the development of his pcf (possible cofinalities) theory. One very quotable result from this theory is the following:

2 ℵ ω < ℵ ω 4 .

In summary, although the continuum function at regular cardinals is relatively unconstrained in ZFC, the continuum function at singular cardinals is (provably in ZFC) constrained in significant ways by the behaviour of the continuum function on the smaller cardinals.

Further Reading : For more cardinal arithmetic see Jech (2003). For more on the case of singular cardinals and pcf theory see Abraham & Magidor (2010) and Holz, Steffens & Weitz (1999).

2. Definable Versions of the Continuum Hypothesis and its Negation

Let us return to the continuum function on regular cardinals and concentrate on the simplest case, the size of 2 ℵ 0 . One of Cantor's original approaches to CH was by investigating “simple” sets of real numbers (see Hallett (1984), pp. 3–5 and §2.3(b)). One of the first results in this direction is the Cantor-Bendixson theorem that every infinite closed set is either countable or contains a perfect subset, in which case it has the same cardinality as the set of reals. In other words, CH holds (in this formulation) when one restricts one's attention to closed sets of reals. In general, questions about “definable” sets of reals are more tractable than questions about arbitrary sets of reals and this suggests looking at definable versions of the continuum hypothesis.

There are three different formulations of the continuum hypothesis—the interpolant version, the well-ordering version, and the surjection version. These versions are all equivalent to one another in ZFC but we shall be imposing a definability constraint and in this case there can be interesting differences (our discussion follows Martin (1976)). There is really a hierarchy of notions of definability—ranging up through the Borel hierarchy, the projective hierarchy, the hierarchy in L (ℝ), and, more generally, the hierarchy of universally Baire sets—and so each of these three general versions is really a hierarchy of versions, each corresponding to a given level of the hierarchy of definability (for a discussion of the hierarchy of definability see §2.2.1 and §4.6 of the entry “ Large Cardinals and Determinacy ”).

2.1.1 Interpolant Version

The first formulation of CH is that there is no interpolant , that is, there is no infinite set A of real numbers such that the cardinality of A is strictly between that of the natural numbers and the real numbers. To obtain definable versions one simply asserts that there is no “definable” interpolant and this leads to a hierarchy of definable interpolant versions, depending on which notion of definability one employs. More precisely, for a given pointclass Γ in the hierarchy of definable sets of reals, the corresponding definable interpolant version of CH asserts that there is no interpolant in Γ.

The Cantor-Bendixson theorem shows that there is no interpolant in Γ in the case where Γ is the pointclass of closed sets, thus verifying this version of CH. This was improved by Suslin who showed that this version of CH holds for Γ where Γ is the class of Σ̰ 1 1 sets. One cannot go much further within ZFC—to prove stronger versions one must bring in stronger assumptions. It turns out that axioms of definable determinacy and large cardinal axioms achieve this. For example, results of Kechris and Martin show that if Δ̰ 1 n -determinacy holds then this version of CH holds for the pointclass of Σ̰ 1 n+1 sets. Going further, if one assumes AD L (ℝ) then this version of CH holds for all sets of real numbers appearing in L (ℝ). Since these hypotheses follow from large cardinal axioms one also has that stronger and stronger large cardinal assumptions secure stronger and stronger versions of this version of the effective continuum hypothesis. Indeed large cardinal axioms imply that this version of CH holds for all sets of reals in the definability hierarchy we are considering; more precisely, if there is a proper class of Woodin cardinals then this version of CH holds for all universally Baire sets of reals.

2.1.2 Well-ordering Version

The second formulation of CH asserts that every well-ordering of the reals has order type less than ℵ 2 . For a given pointclass Γ in the hierarchy, the corresponding definable well-ordering version of CH asserts that every well-ordering (coded by a set) in Γ has order type less than ℵ 2 .

Again, axioms of definable determinacy and large cardinal axioms imply this version of CH for richer notions of definability. For example, if AD L (ℝ) holds then this version of CH holds for all sets of real numbers in L (ℝ). And if there is a proper class of Woodin cardinals then this version of CH holds for all universally Baire sets of reals.

2.1.3 Surjection Version

The third version formulation of CH asserts that there is no surjection ρ : ℝ → ℵ 2 , or, equivalently, that there is no prewellordering of ℝ of length ℵ 2 . For a given pointclass Γ in the hierarchy of definability, the corresponding surjection version of CH asserts that there is no surjection ρ : ℝ → ℵ 2 such that (the code for) ρ is in Γ.

Here the situation is more interesting. Axioms of definable determinacy and large cardinal axioms have bearing on this version since they place bounds on how long definable prewellorderings can be. Let δ̰ 1 n be the supremum of the lengths of the Σ̰ 1 n -prewellorderings of reals and let Θ L (ℝ) be the supremum of the lengths of prewellorderings of reals where the prewellordering is definable in the sense of being in L (ℝ). It is a classical result that δ̰ 1 1 = ℵ 1 . Martin showed that δ̰ 1 2 ≤ ℵ 2 and that if there is a measurable cardinal then δ̰ 1 3 ≤ ℵ 3 . Kunen and Martin also showed under PD, δ̰ 1 4 ≤ ℵ 4 and Jackson showed that under PD, for each n < ω, δ̰ 1 n < ℵ ω . Thus, assuming that there are infinitely many Woodin cardinals, these bounds hold. Moreover, the bounds continue to hold regardless of the size of 2 ℵ 0 . Of course, the question is whether these bounds can be improved to show that the prewellorderings are shorter than ℵ 2 . In 1986 Foreman and Magidor initiated a program to establish this. In the most general form they aimed to show that large cardinal axioms implied that this version of CH held for all universally Baire sets of reals.

2.1.4 Potential Bearing on CH

Notice that in the context of ZFC, these three hierarchies of versions of CH are all successive approximations of CH and in the limit case, where Γ is the pointclass of all sets of reals, they are equivalent to CH. The question is whether these approximations can provide any insight into CH itself.

There is an asymmetry that was pointed out by Martin, namely, that a definable counterexample to CH is a real counterexample, while no matter how far one proceeds in verifying definable versions of CH at no stage will one have touched CH itself. In other words, the definability approach could refute CH but it could not prove it.

Still, one might argue that although the definability approach could not prove CH it might provide some evidence for it. In the case of the first two versions we now know that CH holds for all definable sets. Does this provide evidence of CH? Martin pointed out (before the full results were known) that this is highly doubtful since in each case one is dealing with sets that are atypical. For example, in the first version, at each stage one secures the definable version of CH by showing that all sets in the definability class have the perfect set property; yet such sets are atypical in that assuming AC it is easy to show that there are sets without this property. In the second version, at each stage one actually shows not only that each well-ordering of reals in the definability class has ordertype less than ℵ 2 , but also that it has ordertype less than ℵ 1 . So neither of these versions really illuminates CH.

The third version actually has an advantage in this regard since not all of the sets it deals with are atypical. For example, while all Σ̰ 1 1 -sets have length less than ℵ 1 , there are Π̰ 1 1 -sets of length ℵ 1 . Of course, it could turn out that even if the Foreman-Magidor program were to succeed the sets could turn out to be atypical in another sense, in which case it would shed little light on CH. More interesting, however, is the possibility that in contrast to the first two versions, it would actually provide an actual counterexample to CH. This, of course, would require the failure of the Foreman-Magidor program.

The goal of the Foreman-Magidor program was to show that large cardinal axioms also implied that the third version of CH held for all sets in L (ℝ) and, more generally, all universally Baire sets. In other words, the goal was to show that large cardinal axioms implied that Θ L (ℝ) ≤ ℵ 2 and, more generally, that Θ L (A,ℝ) ≤ ℵ 2 for each universally Baire set A .

The motivation came from the celebrated results of Foreman, Magidor and Shelah on Martin's Maximum (MM), which showed that assuming large cardinal axioms one can always force to obtain a precipitous ideal on ℵ 2 without collapsing ℵ 2 (see Foreman, Magidor & Shelah (1988)). The program involved a two-part strategy:

• Strengthen this result to show that assuming large cardinal axioms one can always force to obtain a saturated ideal on ℵ 2 without collapsing ℵ 2 .
• Show that the existence of such a saturated ideal implies that Θ L (ℝ) ≤ ℵ 2 and, more generally that Θ L (A,ℝ) ≤ ℵ 2 for every universally Baire set A .

This would show that show that Θ L (ℝ) ≤ ℵ 2 and, more generally that Θ L (A,ℝ) ≤ ℵ 2 for every universally Baire set A . [ 4 ]

In December 1991, the following result dashed the hopes of this program.

The point is that the hypothesis of this theorem can always be forced assuming large cardinals. Thus, it is possible to have Θ L (ℝ) > ℵ 2 (in fact, δ̰ 1 3 > ℵ 2 ).

Where did the program go wrong? Foreman and Magidor had an approximation to (B) and in the end it turned out that (B) is true.

So the trouble is with (A).

This illustrates an interesting contrast between our three versions of the effective continuum hypothesis, namely, that they can come apart. For while large cardinals rule out definable counterexamples of the first two kinds, they cannot rule out definable counterexamples of the third kind. But again we must stress that they cannot prove that there are such counterexamples.

But there is an important point: Assuming large cardinal axioms (AD L (ℝ) suffices), although one can produce outer models in which δ̰ 1 3 > ℵ 2 it is not currently known how to produce outer models in which δ̰ 1 3 > ℵ 3 or even Θ L (ℝ) > ℵ 3 . Thus it is an open possibility that from ZFC +AD L (ℝ) one can prove Θ L (ℝ) ≤ ℵ 3 . Were this to be the case, it would follow that although large cardinals cannot rule out the definable failure of CH they can rule out the definable failure of 2 ℵ 0 = ℵ 2 . This could provide some insight into the size of the continuum, underscoring the centrality of ℵ 2 .

Further Reading : For more on the three effective versions of CH see Martin (1976); for more on the Foreman-Magidor program see Foreman & Magidor (1995) and the introduction to Woodin (1999).

3. The Case for ¬CH

The above results led Woodin to the identification of a “canonical” model in which CH fails and this formed the basis of his an argument that CH is false. In Section 3.1 we will describe the model and in the remainder of the section we will present the case for the failure of CH. In Section 3.2 we will introduce Ω-logic and the other notions needed to make the case. In Section 3.3 we will present the case.

The goal is to find a model in which CH is false and which is canonical in the sense that its theory cannot be altered by set forcing in the presence of large cardinals. The background motivation is this: First, we know that in the presence of large cardinal axioms the theory of second-order arithmetic and even the entire theory of L (ℝ) is invariant under set forcing. The importance of this is that it demonstrates that our main independence techniques cannot be used to establish the independence of questions about second-order arithmetic (or about L (ℝ)) in the presence of large cardinals. Second, experience has shown that the large cardinal axioms in question seem to answer all of the major known open problems about second-order arithmetic and L (ℝ) and the set forcing invariance theorems give precise content to the claim that these axioms are “effectively complete”. [ 5 ]

It follows that if ℙ is any homogeneous partial order in L (ℝ) then the generic extension L (ℝ) ℙ inherits the generic absoluteness of L (ℝ). Woodin discovered that there is a very special partial order ℙ max that has this feature. Moreover, the model L (ℝ) ℙ max satisfies ZFC + ¬CH. The key feature of this model is that it is “maximal” (or “saturated”) with respect to sentences that are of a certain complexity and which can be shown to be consistent via set forcing over the model; in other words, if these sentences can hold (by set forcing over the model) then they do hold in the model. To state this more precisely we are going to have to introduce a few rather technical notions.

There are two ways of stratifying the universe of sets. The first is in terms of ⟨ V α | α ∈ On ⟩, the second is in terms of ⟨ H (κ) | κ ∈ Card⟩, where H (κ) is the set of all sets which have cardinality less than κ and whose members have cardinality less than κ, and whose members of members have cardinality less than κ, and so on. For example, H (ω) = V ω and the theories of the structures H (ω 1 ) and V ω+1 are mutually interpretable. This latter structure is the structure of second-order arithmetic and, as mentioned above, large cardinal axioms give us an “effectively complete” understanding of this structure. We should like to be in the same position with regard to larger and larger fragments of the universe and the question is whether we should proceed in terms of the first or the second stratification.

The second stratification is potentially more fine-grained. Assuming CH one has that the theories of H (ω 2 ) and V ω+2 are mutually interpretable and assuming larger and larger fragments of GCH this correspondence continues upward. But if CH is false then the structure H (ω 2 ) is less rich than the structure V ω 2 . In this event the latter structure captures full third-order arithmetic, while the former captures only a small fragment of third-order arithmetic but is nevertheless rich enough to express CH. Given this, in attempting to understand the universe of sets by working up through it level by level, it is sensible to use the potentially more fine-grained stratification.

Our next step is therefore to understand H (ω 2 ). It actually turns out that we will be able to understand slightly more and this is somewhat technical. We will be concerned with the structure ⟨ H (ω 2 ), ∈, I NS , A G ⟩ ⊧ φ, where I NS is the non-stationary ideal on ω 1 and A G is the interpretation of (the canonical representation of) a set of reals A in L (ℝ). The details will not be important and the reader is asked to just think of H (ω 2 ) along with some “extra stuff” and not worry about the details concerning the extra stuff. [ 6 ]

We are now in a position to state the main result:

⟨ H (ω 2 ), ∈, I NS , A G ⟩ ⊧ φ

L (ℝ) ℙ max ⊧ “⟨ H (ω 2 ), ∈, I NS , A⟩ ⊧ φ”.

There are two key points: First, the theory of L (ℝ) ℙ max is “effectively complete” in the sense that it is invariant under set forcing. Second, the model L (ℝ) ℙ max is “maximal” (or “saturated”) in the sense that it satisfies all Π 2 -sentences (about the relevant structure) that can possibly hold (in the sense that they can be shown to be consistent by set forcing over the model).

One would like to get a handle on the theory of this structure by axiomatizing it. The relevant axiom is the following:

Finally, this axiom settles CH:

We will now recast the above results in terms of a strong logic. We shall make full use of large cardinal axioms and in this setting we are interested in logics that are “well-behaved” in the sense that the question of what implies what is not radically independent. For example, it is well known that CH is expressible in full second-order logic. It follows that in the presence of large cardinals one can always use set forcing to flip the truth-value of a purported logical validity of full second-order logic. However, there are strong logics—like ω-logic and β-logic—that do not have this feature—they are well-behaved in the sense that in the presence of large cardinal axioms the question of what implies what cannot be altered by set forcing. We shall introduce a very strong logic that has this feature—Ω-logic. In fact, the logic we shall introduce can be characterized as the strongest logic with this feature (see Koellner (2010) for further discussion of strong logics and for a precise statement of this result).

3.2.1 Ω-logic

T ⊧ Ω φ

if V B α ⊧ T then V B α ⊧ φ.

We say that a statement φ is Ω- satisfiable if there exists an ordinal α and a complete Boolean algebra B such that V B α ⊧ φ, and we say that φ is Ω- valid if ∅ ⊧ Ω φ. So, the above theorem says that (under our background assumptions), the statement “φ is Ω-satisfiable” is generically invariant and in terms of Ω-validity this is simply the following:

T ⊧ Ω φ iff V B ⊧ “T ⊧ Ω φ.”

Thus this logic is robust in that the question of what implies what is invariant under set forcing.

3.2.2 The Ω Conjecture

Corresponding to the semantic relation ⊧ Ω there is a quasi-syntactic proof relation ⊢ Ω . The “proofs” are certain robust sets of reals (universally Baire sets of reals) and the test structures are models that are “closed” under these proofs. The precise notions of “closure” and “proof” are somewhat technical and so we will pass over them in silence. [ 7 ]

Like the semantic relation, this quasi-syntactic proof relation is robust under large cardinal assumptions:

T ⊢ Ω φ iff V B ⊧ ‘T ⊢ Ω φ’.

Thus, we have a semantic consequence relation and a quasi-syntactic proof relation, both of which are robust under the assumption of large cardinal axioms. It is natural to ask whether the soundness and completeness theorems hold for these relations. The soundness theorem is known to hold:

It is open whether the corresponding completeness theorem holds. The Ω Conjecture is simply the assertion that it does:

∅ ⊧ Ω φ iff ∅ ⊢ Ω φ.

We will need a strong form of this conjecture which we shall call the Strong Ω Conjecture. It is somewhat technical and so we will pass over it in silence. [ 8 ]

3.2.3 Ω-Complete Theories

Recall that one key virtue of large cardinal axioms is that they “effectively settle” the theory of second-order arithmetic (and, in fact, the theory of L (ℝ) and more) in the sense that in the presence of large cardinals one cannot use the method of set forcing to establish independence with respect to statements about L (ℝ). This notion of invariance under set forcing played a key role in Section 3.1 . We can now rephrase this notion in terms of Ω-logic.

The invariance of the theory of L (ℝ) under set forcing can now be rephrased as follows:

Unfortunately, it follows from a series of results originating with work of Levy and Solovay that traditional large cardinal axioms do not yield Ω-complete theories at the level of Σ 2 1 since one can always use a “small” (and hence large cardinal preserving) forcing to alter the truth-value of CH.

Nevertheless, if one supplements large cardinal axioms then Ω-complete theories are forthcoming. This is the centerpiece of the case against CH.

• ZFC + A is Ω -satisfiable and
• ZFC + A is Ω -complete for the structure H (ω 2 ) .

ZFC + A ⊧ Ω ‘ H (ω 2 ) ⊧ ¬CH’.

Let us rephrase this as follows: For each A satisfying (1), let

T A = {φ | ZFC + A ⊧ Ω ‘ H (ω 2 ) ⊧ ¬φ’}.

The theorem says that if there is a proper class of Woodin cardinals and the Ω Conjecture holds, then there are (non-trivial) Ω-complete theories T A of H (ω 2 ) and all such theories contain ¬CH.

It is natural to ask whether there is greater agreement among the Ω-complete theories T A . Ideally, there would be just one. A recent result (building on Theorem 5.5) shows that if there is one such theory then there are many such theories.

 i.  ZFC + A is Ω -satisfiable and ii.  ZFC + A is Ω -complete for the structure H (ω 2 ) .

 i′.  ZFC + B is Ω -satisfiable and ii′.  ZFC + B is Ω -complete for the structure H (ω 2 )

How then shall one select from among these theories? Woodin's work in this area goes a good deal beyond Theorem 5.1. In addition to isolating an axiom that satisfies (1) of Theorem 5.1 (assuming Ω-satisfiability), he isolates a very special such axiom, namely, the axiom (∗) (“star”) mentioned earlier.

This axiom can be phrased in terms of (the provability notion of) Ω-logic:

• (∗) .

⟨ H (ω 2 ), ∈, I NS , A | A ∈ 𝒫 (ℝ) ∩ L (ℝ)⟩

ZFC + “⟨ H (ω 2 ), ∈, I NS , A | A ∈ 𝒫 (ℝ) ∩ L (ℝ)⟩ ⊧ φ”

⟨ H (ω 2 ), ∈, I NS , A | A ∈ 𝒫 (ℝ) ∩ L (ℝ)⟩ ⊧ φ.

It follows that of the various theories T A involved in Theorem 5.1, there is one that stands out: The theory T (∗) given by (∗). This theory maximizes the Π 2 -theory of the structure ⟨ H (ω 2 ), ∈, I NS , A | A ∈ 𝒫 (ℝ) ∩ L (ℝ)⟩.

The continuum hypothesis fails in this theory. Moreover, in the maximal theory T (∗) given by (∗) the size of the continuum is ℵ 2 . [ 9 ]

To summarize: Assuming the Strong Ω Conjecture, there is a “good” theory of H (ω 2 ) and all such theories imply that CH fails. Moreover, (again, assuming the Strong Ω Conjecture) there is a maximal such theory and in that theory 2 ℵ 0 = ℵ 2 .

Further Reading : For the mathematics concerning ℙ max see Woodin (1999). For an introduction to Ω-logic see Bagaria, Castells & Larson (2006). For more on incompatible Ω-complete theories see Koellner & Woodin (2009). For more on the case against CH see Woodin (2001a,b, 2005a,b).

4. The Multiverse

The above case for the failure of CH is the strongest known local case for axioms that settle CH. In this section and the next we will switch sides and consider the pluralist arguments to the effect that CH does not have an answer (in this section) and to the effect that there is an equally good case for CH (in the next section). In the final two section we will investigate optimistic global scenarios that provide hope of settling the issue.

The pluralist maintains that the independence results effectively settle the undecided questions by showing that they have no answer. One way of providing a foundational framework for such a view is in terms of the multiverse. On this view there is not a single universe of set theory but rather a multiverse of legitimate candidates, some of which may be preferable to others for certain purposes but none of which can be said to be the “true” universe. The multiverse conception of truth is the view that a statement of set theory can only be said to be true simpliciter if it is true in all universes of the multiverse. For the purposes of this discussion we shall say that a statement is indeterminate according to the multiverse conception if it is neither true nor false according to the multiverse conception. How radical such a view is depends on the breadth of the conception of the multiverse.

The pluralist is generally a non-pluralist about certain domains of mathematics. For example, a strict finitist might be a non-pluralist about PA but a pluralist about set theory and one might be a non-pluralist about ZFC and a pluralist about large cardinal axioms and statements like CH.

There is a form of radical pluralism which advocates pluralism concerning all domains of mathematics. On this view any consistent theory is a legitimate candidate and the corresponding models of such theories are legitimate candidates for the domain of mathematics. Let us call this the broadest multiverse view. There is a difficulty in articulating this view, which may be brought out as follows: To begin with, one must pick a background theory in which to discuss the various models and this leads to a difficult. For example, according to the broad multiverse conception, since PA cannot prove Con(PA) (by the second incompleteness theorem, assuming that PA is consistent) there are models of PA + ¬Con(PA) and these models are legitimate candidates, that is, they are universes within the broad multiverse. Now to arrive at this conclusion one must (in the background theory) be in a position to prove Con(PA) (since this assumption is required to apply the second incompleteness theorem in this particular case). Thus, from the perspective of the background theory used to argue that the above models are legitimate candidates, the models in question satisfy a false Σ 0 1 -sentence, namely, ¬Con(PA). In short, there is a lack of harmony between what is held at the meta-level and what is held at the object-level.

The only way out of this difficulty would seem to be to regard each viewpoint—each articulation of the multiverse conception—as provisional and, when pressed, embrace pluralism concerning the background theory. In other words, one would have to adopt a multiverse conception of the multiverse, a multiverse conception of the multiverse conception of the multiverse, and so on, off to infinity. It follows that such a position can never be fully articulated—each time one attempts to articulate the broad multiverse conception one must employ a background theory but since one is a pluralist about that background theory this pass at using the broad multiverse to articulate the conception does not do the conception full justice. The position is thus difficult to articulate. One can certainly take the pluralist stance and try to gesture toward or exhibit the view that one intends by provisionally settling on a particular background theory but then advocate pluralism regarding that when pressed. The view is thus something of a “moving target”. We shall pass over this view in silence and concentrate on views that can be articulated within a foundational framework.

We will accordingly look at views which embrace non-pluralism with regard to a given stretch of mathematics and for reasons of space and because this is an entry on set theory we will pass over the long debates concerning strict finitism, finitism, predicativism, and start with views that embrace non-pluralism regarding ZFC.

Let the broad multiverse (based on ZFC) be the collection of all models of ZFC. The broad multiverse conception of truth (based on ZFC) is then simply the view that a statement of set theory is true simpliciter if it is provable in ZFC. On this view the statement Con(ZFC) and other undecided Π 0 1 -statements are classified as indeterminate. This view thus faces a difficulty parallel to the one mentioned above concerning radical pluralism.

This motivates the shift to views that narrow the class of universes in the multiverse by employing a strong logic. For example, one can restrict to universes that are ω-models, β-models (i.e., wellfounded), etc. On the view where one takes ω-models, the statement Con(ZFC) is classified as true (though this is sensitive to the background theory) but the statement PM (all projective sets are Lebesgue measurable) is classified as indeterminate.

For those who are convinced by the arguments (surveyed in the entry “ Large Cardinals and Determinacy ”) for large cardinal axioms and axioms of definable determinacy, even these multiverse conceptions are too weak. We will follow this route. For the rest of this entry we will embrace non-pluralism concerning large cardinal axioms and axioms of definable determinacy and focus on the question of CH.

The motivation behind the generic multiverse is to grant the case for large cardinal axioms and definable determinacy but deny that statements such as CH have a determinate truth value. To be specific about the background theory let us take ZFC + “There is a proper class of Woodin cardinals” and recall that this large cardinal assumption secures axioms of definable determinacy such as PD and AD L (ℝ) .

Let the generic multiverse 𝕍 be the result of closing V under generic extensions and generic refinements. One way to formalize this is by taking an external vantage point and start with a countable transitive model M . The generic multiverse based on M is then the smallest set 𝕍 M such that M ∈ 𝕍 M and, for each pair of countable transitive models ( N , N [ G ]) such that N ⊧ ZFC and G ⊆ ℙ is N -generic for some partial order in ℙ ∈ N , if either N or N [ G ] is in 𝕍 M then both N and N [ G ] are in 𝕍 M .

Let the generic multiverse conception of truth be the view that a statement is true simpliciter iff it is true in all universes of the generic multiverse. We will call such a statement a generic multiverse truth . A statement is said to be indeterminate according to the generic multiverse conception iff it is neither true nor false according to the generic multiverse conception. For example, granting our large cardinal assumptions, such a view deems PM (and PD and AD L (ℝ) ) true but deems CH indeterminate.

Is the generic multiverse conception of truth tenable? The answer to this question is closely related to the subject of Ω-logic. The basic connection between generic multiverse truth and Ω-logic is embodied in the following theorem:

• φ is a generic multiverse truth.
• φ is Ω -valid.

Now, recall that by Theorem 3.5, under our background assumptions, Ω-validity is generically invariant. It follows that given our background theory, the notion of generic multiverse truth is robust with respect to Π 2 -statements. In particular, for Π 2 -statements, the statement “φ is indeterminate” is itself determinate according to the generic multiverse conception. In this sense the conception of truth is not “self-undermining” and one is not sent in a downward spiral where one has to countenance multiverses of multiverses. So it passes the first test. Whether it passes a more challenging test depends on the Ω Conjecture.

The Ω Conjecture has profound consequences for the generic multiverse conception of truth. Let

𝒱 Ω = {φ | ∅ ⊧ Ω φ}

and, for any specifiable cardinal κ, let

𝒱 Ω ( H (κ + )) = {φ | ZFC ⊧ Ω “ H (κ + ) ⊧ φ”},

where recall that H (κ + ) is the collection of sets of hereditary cardinality less than κ + . Thus, assuming ZFC and that there is a proper class of Woodin cardinals, the set 𝒱 Ω is Turing equivalent to the set of Π 2 generic multiverse truths and the set 𝒱 Ω ( H (κ + )) is precisely the set of generic multiverse truths of H (κ + ).

To describe the bearing of the Ω Conjecture on the generic-multiverse conception of truth, we introduce two Transcendence Principles which serve as constraints on any tenable conception of truth in set theory—a truth constraint and a definability constraint .

This constraint is in the spirit of those principles of set theory—most notably, reflection principles—which aim to capture the pretheoretic idea that the universe of sets is so rich that it cannot “be described from below”; more precisely, it asserts that any tenable conception of truth must respect the idea that the universe of sets is so rich that truth (or even just Π 2 -truth) cannot be described in some specifiable fragment. (Notice that by Tarski's theorem on the undefinability of truth, the truth constraint is trivially satisfied by the standard conception of truth in set theory which takes the multiverse to contain a single element, namely, V .)

There is also a related constraint concerning the definability of truth. For a specifiable cardinal κ, set Y ⊆ ω is definable in H (κ + ) across the multiverse if Y is definable in the structure H (κ + ) of each universe of the multiverse (possibly by formulas which depend on the parent universe).

Notice again that by Tarski's theorem on the undefinability of truth, the definability constraint is trivially satisfied by the degenerate multiverse conception that takes the multiverse to contain the single element V . (Notice also that if one modifies the definability constraint by adding the requirement that the definition be uniform across the multiverse, then the constraint would automatically be met.)

The bearing of the Ω Conjecture on the tenability of the generic-multiverse conception of truth is contained in the following two theorems:

In other words, if there is a proper class of Woodin cardinals and if the Ω Conjecture holds then the generic multiverse conception of truth violates both the Truth Constraint (at δ 0 ) and the Definability Constraint (at δ 0 ).

There are actually sharper versions of the above results that involve H ( c + ) in place of H (δ + 0 ).

In other words, if there is a proper class of Woodin cardinals and if the Ω Conjecture holds then the generic-multiverse conception of truth violates the Truth Constraint at the level of third-order arithmetic, and if, in addition, the AD + Conjecture holds, then the generic-multiverse conception of truth violates the Definability Constraint at the level of third-order arithmetic.

There appear to be four ways that the advocate of the generic multiverse might resist the above criticism.

First, one could maintain that the Ω Conjecture is just as problematic as CH and hence like CH it is to be regarded as indeterminate according to the generic-multiverse conception of truth. The difficulty with this approach is the following:

V ⊧ Ω-conjecture iff V 𝔹 ⊧ Ω-conjecture.

Thus, in contrast to CH, the Ω Conjecture cannot be shown to be independent of ZFC + “There is a proper class of Woodin cardinals” via set forcing. In terms of the generic multiverse conception of truth, we can put the point this way: While the generic-multiverse conception of truth deems CH to be indeterminate, it does not deem the Ω Conjecture to be indeterminate. So the above response is not available to the advocate of the generic-multiverse conception of truth. The advocate of that conception already deems the Ω Conjecture to be determinate.

Second, one could grant that the Ω Conjecture is determinate but maintain that it is false. There are ways in which one might do this but that does not undercut the above argument. The reason is the following: To begin with there is a closely related Σ 2 -statement that one can substitute for the Ω Conjecture in the above arguments. This is the statement that the Ω Conjecture is (non-trivially) Ω-satisfiable, that is, the statement: There exists an ordinal α and a universe V′ of the multiverse such that

V′ α ⊧ ZFC + “There is a proper class of Woodin cardinals”

V′ α ⊧ “The Ω Conjecture”.

This Σ 2 -statement is invariant under set forcing and hence is one adherents to the generic multiverse view of truth must deem determinate. Moreover, the key arguments above go through with this Σ 2 -statement instead of the Ω Conjecture. The person taking this second line of response would thus also have to maintain that this statement is false. But there is substantial evidence that this statement is true . The reason is that there is no known example of a Σ 2 -statement that is invariant under set forcing relative to large cardinal axioms and which cannot be settled by large cardinal axioms. (Such a statement would be a candidate for an absolutely undecidable statement.) So it is reasonable to expect that this statement is resolved by large cardinal axioms. However, recent advances in inner model theory—in particular, those in Woodin (2010)—provide evidence that no large cardinal axiom can refute this statement. Putting everything together: It is very likely that this statement is in fact true ; so this line of response is not promising.

Third, one could reject either the Truth Constraint or the Definability Constraint. The trouble is that if one rejects the Truth Constraint then on this view (assuming the Ω Conjecture) Π 2 truth in set theory is reducible in the sense of Turing reducibility to truth in H (δ 0 ) (or, assuming the Strong Ω Conjecture, H ( c + )). And if one rejects the Definability Constraint then on this view (assuming the Ω Conjecture) Π 2 truth in set theory is reducible in the sense of definability to truth in H (δ 0 ) (or, assuming the Strong Ω Conjecture, H ( c + )). On either view, the reduction is in tension with the acceptance of non-pluralism regarding the background theory ZFC + “There is a proper class of Woodin cardinals”.

Fourth, one could embrace the criticism, reject the generic multiverse conception of truth, and admit that there are some statements about H (δ + 0 ) (or H ( c + ), granting, in addition, the AD + Conjecture) that are true simpliciter but not true in the sense of the generic-multiverse, and yet nevertheless continue to maintain that CH is indeterminate. The difficulty is that any such sentence φ is qualitatively just like CH in that it can be forced to hold and forced to fail. The challenge for the advocate of this approach is to modify the generic-multiverse conception of truth in such a way that it counts φ as determinate and yet counts CH as indeterminate.

In summary: There is evidence that the only way out is the fourth way out and this places the burden back on the pluralist—the pluralist must come up with a modified version of the generic multiverse.

Further Reading : For more on the connection between Ω-logic and the generic multiverse and the above criticism of the generic multiverse see Woodin (2011a). For the bearing of recent results in inner model theory on the status of the Ω Conjecture see Woodin (2010).

5. The Local Case Revisited

Let us now turn to a second way in which one might resist the local case for the failure of CH. This involves a parallel case for CH. In Section 5.1 we will review the main features of the case for ¬CH in order to compare it with the parallel case for CH. In Section 5.2 we will present the parallel case for CH. In Section 5.3 we will assess the comparison.

Recall that there are two basic steps in the case presented in Section 3.3 . The first step involves Ω-completeness (and this gives ¬CH) and the second step involves maximality (and this gives the stronger 2 ℵ 0 = ℵ 2 ). For ease of comparison we shall repeat these features here:

The first step is based on the following result:

ZFC + A ⊧ Ω “ H (ω 2 ) ⊧ ¬CH”.

T A = {φ | ZFC + A ⊧ Ω “ H (ω 2 ) ⊧ ¬φ”}.

The theorem says that if there is a proper class of Woodin cardinals and the Strong Ω Conjecture holds, then there are (non-trivial) Ω-complete theories T A of H (ω 2 ) and all such theories contain ¬CH. In other words, under these assumptions, there is a “good” theory and all “good” theories imply ¬CH.

The second step begins with the question of whether there is greater agreement among the Ω-complete theories T A . Ideally, there would be just one. However, this is not the case.

Then there is an axiom B such that

This raises the issue as to how one is to select from among these theories? It turns out that there is a maximal theory among the T A and this is given by the axiom (∗).

is Ω -consistent, then

So, of the various theories T A involved in Theorem 5.1, there is one that stands out: The theory T (∗) given by (∗). This theory maximizes the Π 2 -theory of the structure ⟨ H (ω 2 ), ∈, I NS , A | A ∈ 𝒫 (ℝ) ∩ L (ℝ)⟩. The fundamental result is that in this maximal theory

2 ℵ 0 = ℵ 2 .

The parallel case for CH also has two steps, the first involving Ω-completeness and the second involving maximality.

The first result in the first step is the following:

Moreover, up to Ω-equivalence, CH is the unique Σ 2 1 -statement that is Ω-complete for Σ 2 1 ; that is, letting T A be the Ω-complete theory given by ZFC + A where A is Σ 2 1 , all such T A are Ω-equivalent to T CH and hence (trivially) all such T A contain CH. In other words, there is a “good” theory and all “good” theories imply CH.

To complete the first step we have to determine whether this result is robust. For it could be the case that when one considers the next level, Σ 2 2 (or further levels, like third-order arithmetic) CH is no longer part of the picture, that is, perhaps large cardinals imply that there is an axiom A such that ZFC + A is Ω-complete for Σ 2 2 (or, going further, all of third order arithmetic) and yet not all such A have an associated T A which contains CH. We must rule this out if we are to secure the first step.

The most optimistic scenario along these lines is this: The scenario is that there is a large cardinal axiom L and axioms A → such that ZFC + L + A → is Ω-complete for all of third-order arithmetic and all such theories are Ω-equivalent and imply CH. Going further, perhaps for each specifiable fragment V λ of the universe of sets there is a large cardinal axiom L and axioms A → such that ZFC + L + A → is Ω-complete for the entire theory of V λ and, moreover, that such theories are Ω-equivalent and imply CH. Were this to be the case it would mean that for each such λ there is a unique Ω-complete picture of V λ and we would have a unique Ω-complete understanding of arbitrarily large fragments of the universe of sets. This would make for a strong case for new axioms completing the axioms of ZFC and large cardinal axioms.

Unfortunately, this optimistic scenario fails: Assuming the existence of one such theory one can construct another which differs on CH:

ZFC + L + A → is Ω-complete for Th( V λ ).

ZFC + L + B → is Ω-complete for Th( V λ )

This still leaves us with the question of existence and the answer to this question is sensitive to the Ω Conjecture and the AD + Conjecture:

In fact, under a stronger assumption, the scenario must fail at a much earlier level.

It is open whether there can be such a theory at the level of Σ 2 2 . It is conjectured that ZFC + ◇ is Ω-complete (assuming large cardinal axioms) for Σ 2 2 .

Let us assume that it is answered positively and return to the question of uniqueness. For each such axiom A , let T A be the Σ 2 2 theory computed by ZFC + A in Ω-logic. The question of uniqueness simply asks whether T A is unique.

 i. ZFC + A is Ω -satisfiable and ii. ZFC + A is Ω -complete for Σ 2 2 .

 i′. ZFC + B is Ω -satisfiable and ii′. ZFC + B is Ω -complete for Σ 2 2

This is the parallel of Theorem 5.2.

To complete the parallel one would need that CH is among all of the T A . This is not known. But it is a reasonable conjecture.

• ZFC + A is Ω-satisfiable and
• ZFC + A is Ω-complete for the Σ 2 2 .

ZFC + A ⊧ Ω CH.

Should this conjecture hold it would provide a true analogue of Theorem 5.1. This would complete the parallel with the first step.

There is also a parallel with the second step. Recall that for the second step in the previous subsection we had that although the various T A did not agree, they all contained ¬CH and, moreover, from among them there is one that stands out, namely the theory given by (∗), since this theory maximizes the Π 2 -theory of the structure ⟨ H (ω 2 ), ∈, I NS , A | A ∈ P (ℝ) ∩ L (ℝ)⟩. In the present context of CH we again (assuming the conjecture) have that although the T A do not agree, they all contain CH. It turns out that once again, from among them there is one that stands out, namely, the maximum one. For it is known (by a result of Woodin in 1985) that if there is a proper class of measurable Woodin cardinals then there is a forcing extension satisfying all Σ 2 2 sentences φ such that ZFC + CH + φ is Ω-satisfiable (see Ketchersid, Larson, & Zapletal (2010)). It follows that if the question of existence is answered positively with an A that is Σ 2 2 then T A must be this maximum Σ 2 2 theory and, consequently, all T A agree when A is Σ 2 2 . So, assuming that there is a T A where A is Σ 2 2 , then, although not all T A agree (when A is arbitrary) there is one that stands out, namely, the one that is maximum for Σ 2 2 sentences.

Thus, if the above conjecture holds, then the case of CH parallels that of ¬CH, only now Σ 2 2 takes the place of the theory of H (ω 2 ).

Assuming that the conjecture holds the case of CH parallels that of ¬CH, only now Σ 2 2 takes the place of the theory of H (ω 2 ): Under the background assumptions we have:

• there are A such that ZFC + A is Ω-complete for H (ω 2 )
• for every such A the associated T A contains ¬CH, and
• there is a T A which is maximal, namely, T (∗) and this theory contains 2 ℵ 0 = ℵ 2 .
• there are Σ 2 2 -axioms A such that ZFC + A is Ω-complete for Σ 2 2
• for every such A the associated T A contains CH, and
• there is a T A which is maximal.

The two situations are parallel with regard to maximality but in terms of the level of Ω-completeness the first is stronger. For in the first case we are not just getting Ω-completeness with regard to the Π 2 theory of H (ω 2 ) (with the additional predicates), rather we are getting Ω-completeness with regard to all of H (ω 2 ). This is arguably an argument in favour of the case for ¬CH, even granting the conjecture.

But there is a stronger point. There is evidence coming from inner model theory (which we shall discuss in the next section) to the effect that the conjecture is in fact false . Should this turn out to be the case it would break the parallel, strengthening the case for ¬CH.

However, one might counter this as follows: The higher degree of Ω-completeness in the case for ¬CH is really illusory since it is an artifact of the fact that under (∗) the theory of H (ω 2 ) is in fact mutually interpretable with that of H (ω 1 ) (by a deep result of Woodin). Moreover, this latter fact is in conflict with the spirit of the Transcendence Principles discussed in Section 4.3 . Those principles were invoked in an argument to the effect that CH does not have an answer. Thus, when all the dust settles the real import of Woodin's work on CH (so the argument goes) is not that CH is false but rather that CH very likely has an answer.

It seems fair to say that at this stage the status of the local approaches to resolving CH is somewhat unsettled. For this reason, in the remainder of this entry we shall focus on global approaches to settling CH. We shall very briefly discuss two such approaches—the approach via inner model theory and the approach via quasi-large cardinal axioms.

6. The Ultimate Inner Model

Inner model theory aims to produce “ L -like” models that contain large cardinal axioms. For each large cardinal axiom Φ that has been reached by inner model theory, one has an axiom of the form V = L Φ . This axiom has the virtue that (just as in the simplest case of V = L ) it provides an “effectively complete” solution regarding questions about L Φ (which, by assumption, is V ). Unfortunately, it turns out that the axiom V = L Φ is incompatible with stronger large cardinal axioms Φ'. For this reason, axioms of this form have never been considered as plausible candidates for new axioms.

But recent developments in inner model theory (due to Woodin) show that everything changes at the level of a supercompact cardinal. These developments show that if there is an inner model N which “inherits” a supercompact cardinal from V (in the manner in which one would expect, given the trajectory of inner model theory), then there are two remarkable consequences: First, N is close to V (in, for example, the sense that for sufficiently large singular cardinals λ, N correctly computes λ + ). Second, N inherits all known large cardinals that exist in V . Thus, in contrast to the inner models that have been developed thus far, an inner model at the level of a supercompact would provide one with an axiom that could not be refuted by stronger large cardinal assumptions.

The issue, of course, is whether one can have an “ L -like” model (one that yields an “effectively complete” axiom) at this level. There is reason to believe that one can. There is now a candidate model L Ω that yields an axiom V = L Ω with the following features: First, V = L Ω is “effectively complete.” Second, V = L Ω is compatible with all large cardinal axioms. Thus, on this scenario, the ultimate theory would be the (open-ended) theory ZFC + V = L Ω + LCA, where LCA is a schema standing for “large cardinal axioms.” The large cardinal axioms will catch instances of Gödelian independence and the axiom V = L Ω will capture the remaining instances of independence. This theory would imply CH and settle the remaining undecided statements. Independence would cease to be an issue.

It turns out, however, that there are other candidate axioms that share these features, and so the spectre of pluralism reappears. For example, there are axioms V = L Ω S and V = L Ω (∗) . These axioms would also be “effectively complete” and compatible with all large cardinal axioms. Yet they would resolve various questions differently than the axiom V = L Ω . For example, the axiom, V = L Ω (∗) would imply ¬CH. How, then, is one to adjudicate between them?

Further Reading : For an introduction to inner model theory see Mitchell (2010) and Steel (2010). For more on the recent developments at the level of one supercompact and beyond see Woodin (2010).

7. The Structure Theory of L ( V λ+1 )

This brings us to the second global approach, one that promises to select the correct axiom from among V = L Ω , V = L Ω S , V = L Ω (∗) , and their variants. This approach is based on the remarkable analogy between the structure theory of L (ℝ) under the assumption of AD L (ℝ) and the structure theory of L ( V λ+1 ) under the assumption that there is an elementary embedding from L ( V λ+1 ) into itself with critical point below λ. This embedding assumption is the strongest large cardinal axiom that appears in the literature.

The analogy between L (ℝ) and L ( V λ+1 ) is based on the observation that L (ℝ) is simply L ( V ω+1 ). Thus, λ is the analogue of ω, λ + is the analogue of ω 1 , and so on. As an example of the parallel between the structure theory of L (ℝ) under AD L (ℝ) and the structure theory of L ( V λ+1 ) under the embedding axiom, let us mention that in the first case, ω 1 is a measurable cardinal in L (ℝ) and, in the second case, the analogue of ω 1 —namely, λ + —is a measurable cardinal in L ( V λ+1 ). This result is due to Woodin and is just one instance from among many examples of the parallel that are contained in his work.

Now, we have a great deal of information about the structure theory of L (ℝ) under AD L (ℝ) . Indeed, as we noted above, this axiom is “effectively complete” with regard to questions about L (ℝ). In contrast, the embedding axiom on its own is not sufficient to imply that L ( V λ+1 ) has a structure theory that fully parallels that of L (ℝ) under AD L (ℝ) . However, the existence of an already rich parallel is evidence that the parallel extends, and we can supplement the embedding axiom by adding some key components. When one does so, something remarkable happens: the supplementary axioms become forcing fragile . This means that they have the potential to erase independence and provide non-trivial information about V λ+1 . For example, these supplementary axioms might settle CH and much more.

The difficulty in investigating the possibilities for the structure theory of L ( V λ+1 ) is that we have not had the proper lenses through which to view it. The trouble is that the model L ( V λ+1 ) contains a large piece of the universe—namely, L ( V λ+1 )—and the theory of this structure is radically underdetermined. The results discussed above provide us with the proper lenses. For one can examine the structure theory of L ( V λ+1 ) in the context of ultimate inner models like L Ω , L Ω S , L Ω (∗) , and their variants. The point is that these models can accommodate the embedding axiom and, within each, one will be able to compute the structure theory of L ( V λ+1 ).

This provides a means to select the correct axiom from among V = L Ω , V = L Ω S , V = L Ω (∗) , and their variants. One simply looks at the L ( V λ+1 ) of each model (where the embedding axiom holds) and checks to see which has the true analogue of the structure theory of L (ℝ) under the assumption of AD L (ℝ) . It is already known that certain pieces of the structure theory cannot hold in L Ω . But it is open whether they can hold in L Ω S .

Let us consider one such (very optimistic) scenario: The true analogue of the structure theory of L (ℝ) under AD L (ℝ) holds of the L ( V λ+1 ) of L Ω S but not of any of its variants. Moreover, this structure theory is “effectively complete” for the theory of V λ+1 . Assuming that there is a proper class of λ where the embedding axiom holds, this gives an “effectively complete” theory of V . And, remarkably, part of that theory is that V must be L Ω S . This (admittedly very optimistic) scenario would constitute a very strong case for axioms that resolve all of the undecided statements.

One should not place too much weight on this particular scenario. It is just one of many. The point is that we are now in a position to write down a list of definite questions with the following features: First, the questions on this list will have answers—independence is not an issue. Second, if the answers converge then one will have strong evidence for new axioms settling the undecided statements (and hence non-pluralism about the universe of sets); while if the answers oscillate, one will have evidence that these statements are “absolutely undecidable” and this will strengthen the case for pluralism. In this way the questions of “absolute undecidability” and pluralism are given mathematical traction.

Further Reading : For more on the structure theory of L ( V λ+1 ) and the parallel with determinacy see Woodin (2011b).

• Abraham, U. and M. Magidor, 2010, “Cardinal arithmetic,” in Foreman and Kanamori 2010.
• Bagaria, J., N. Castells, and P. Larson, 2006, “An Ω-logic primer,” in J. Bagaria and S. Todorcevic (eds), Set theory , Trends in Mathematics, Birkhäuser, Basel, pp. 1–28.
• Cohen, P., 1963, “The independence of the continuum hypothesis I,” Proceedings of the U.S. National Academy of Sciemces , 50: 1143–48.
• Foreman, M. and A. Kanamori, 2010, Handbook of Set Theory , Springer-Verlag.
• Foreman, M. and M. Magidor, 1995, “Large cardinals and definable counterexamples to the continuum hypothesis,” Annals of Pure and Applied Logic 76: 47–97.
• Foreman, M., M. Magidor, and S. Shelah, 1988, “Martin's Maximum, saturated ideals, and non-regular ultrafilters. Part i,” Annals of Mathematics 127: 1–47.
• Gödel, K., 1938a. “The consistency of the axiom of choice and of the generalized continuum-hypothesis,” Proceedings of the U.S. National Academy of Sciences , 24: 556–7.
• Gödel, K., 1938b. “Consistency-proof for the generalized continuum-hypothesis,” Proceedings of the U.S. National Academy of Sciemces , 25: 220–4.
• Hallett, M., 1984, Cantorian Set Theory and Limitation of Size , Vol. 10 of Oxford Logic Guides , Oxford University Press.
• Holz, M., K. Steffens, and E. Weitz, 1999, Introduction to Cardinal Arithmetic , Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel.
• Jech, T. J., 2003, Set Theory: Third Millennium Edition, Revised and Expanded , Springer-Verlag, Berlin.
• Ketchersid, R., P. Larson, and J. Zapletal, 2010, “Regular embeddings of the stationary tower and Woodin's Sigma-2-2 maximality theorem.” Journal of Symbolic Logic 75(2):711–727.
• Koellner, P., 2010, “Strong logics of first and second order,” Bulletin of Symbolic Logic 16(1): 1–36.
• Koellner, P. and W. H. Woodin, 2009, “Incompatible Ω-complete theories,” The Journal of Symbolic Logic 74 (4).
• Martin, D. A., 1976, “Hilbert's first problem: The Continuum Hypothesis,” in F. Browder (ed.), Mathematical Developments Arising from Hilbert's Problems , Vol. 28 of Proceedings of Symposia in Pure Mathematics , American Mathematical Society, Providence, pp. 81–92.
• Mitchell, W., 2010, “Beginning inner model theory,” in Foreman and Kanamori 2010.
• Steel, J. R., 2010, “An outline of inner model theory,” in Foreman and Kanamori 2010.
• Woodin, W. H., 1999, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal , Vol. 1 of de Gruyter Series in Logic and its Applications , de Gruyter, Berlin.
• –––, 2001a, “The continuum hypothesis, part I,” Notices of the American Mathematical Society 48(6): 567–576.
• –––, 2001b, “The continuum hypothesis, part II,” Notices of the American Mathematical Society 48(7): 681–690.
• –––, 2005a, “The continuum hypothesis,” in R. Cori, A. Razborov, S. Todorĉević and C. Wood (eds), Logic Colloquium 2000 , Vol. 19 of Lecture Notes in Logic , Association of Symbolic Logic, pp. 143–197.
• –––, 2005b, “Set theory after Russell: the journey back to Eden,” in G. Link (ed.), One Hundred Years Of Russell's Paradox: Mathematics, Logic, Philosophy , Vol. 6 of de Gruyter Series in Logic and Its Applications , Walter De Gruyter Inc, pp. 29–47.
• –––, 2010, “Suitable extender models I,” Journal of Mathematical Logic 10(1–2): 101–339.
• –––, 2011a, “The Continuum Hypothesis, the generic-multiverse of sets, and the Ω-conjecture,” in J. Kennedy and R. Kossak, (eds), Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies , Vol. 36 of Lecture Notes in Logic , Cambridge University Press.
• –––, 2011b, “Suitable extender models II,” Journal of Mathematical Logic 11(2): 115–436.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

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continuum hypothesis

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• HMC Mathematics Online Tutorial - Continuum Hypothesis
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continuum hypothesis , statement of set theory that the set of real number s (the continuum) is in a sense as small as it can be. In 1873 the German mathematician Georg Cantor proved that the continuum is uncountable—that is, the real numbers are a larger infinity than the counting numbers—a key result in starting set theory as a mathematical subject. Furthermore, Cantor developed a way of classifying the size of infinite sets according to the number of its elements, or its cardinality . ( See set theory: Cardinality and transfinite numbers .) In these terms, the continuum hypothesis can be stated as follows: The cardinality of the continuum is the smallest uncountable cardinal number.

In Cantor’s notation, the continuum hypothesis can be stated by the simple equation 2 ℵ 0  = ℵ 1 , where ℵ 0 is the cardinal number of an infinite countable set (such as the set of natural numbers), and the cardinal numbers of larger “ well-orderable sets ” are ℵ 1 , ℵ 2 , …, ℵ α , …, indexed by the ordinal numbers. The cardinality of the continuum can be shown to equal 2 ℵ 0 ; thus, the continuum hypothesis rules out the existence of a set of size intermediate between the natural numbers and the continuum.

A stronger statement is the generalized continuum hypothesis (GCH): 2 ℵ α  = ℵ α + 1 for each ordinal number α. The Polish mathematician Wacław Sierpiński proved that with GCH one can derive the axiom of choice .

Since ZF neither proves nor disproves the continuum hypothesis, there remains the question of whether to accept the continuum hypothesis based on an informal concept of what sets are. The general answer in the mathematical community has been negative: the continuum hypothesis is a limiting statement in a context where there is no known reason to impose a limit. In set theory, the power-set operation assigns to each set of cardinality ℵ α its set of all subsets, which has cardinality 2 ℵ α . There seems to be no reason to impose a limit on the variety of subsets that an infinite set might have.

Continuum Hypothesis

Gödel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory . However, using a technique called forcing , Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory . Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the axiom of choice ).

Woodin (2001ab, 2002) formulated a new plausible "axiom" whose adoption (in addition to the Zermelo-Fraenkel axioms and axiom of choice ) would imply that the continuum hypothesis is false. Since set theoreticians have felt for some time that the Continuum Hypothesis should be false, if Woodin's axiom proves to be particularly elegant, useful, or intuitive, it may catch on. It is interesting to compare this to a situation with Euclid's parallel postulate more than 300 years ago, when Wallis proposed an additional axiom that would imply the parallel postulate (Greenberg 1994, pp. 152-153).

Portions of this entry contributed by Matthew Szudzik

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Continuum hypothesis

The hypothesis, due to G. Cantor (1878), stating that every infinite subset of the continuum $\mathbf{R}$ is either equivalent to the set of natural numbers or to $\mathbf{R}$ itself. An equivalent formulation (in the presence of the axiom of choice ) is: $$ 2^{\aleph_0} = \aleph_1 $$ (see Aleph ). The generalization of this equality to arbitrary cardinal numbers is called the generalized continuum hypothesis (GCH): For every ordinal number $\alpha$, \begin{equation} \label{eq:1} 2^{\aleph_\alpha} = \aleph_{\alpha+1} \ . \end{equation}

In the absence of the axiom of choice, the generalized continuum hypothesis is stated in the form \begin{equation} \label{eq:2} \forall \mathfrak{k} \,\,\neg \exists \mathfrak{m}\ (\,\mathfrak{k} < \mathfrak{m} < 2^{\mathfrak{k}}\,) \end{equation} where $\mathfrak{k}$,$\mathfrak{m}$ stand for infinite cardinal numbers. The axiom of choice and (1) follow from (2), while (1) and the axiom of choice together imply (2).

D. Hilbert posed, in his celebrated list of problems, as Problem 1 that of proving Cantor's continuum hypothesis (the problem of the continuum). This problem did not yield a solution within the framework of traditional set-theoretical methods of solution. Among mathematicians the conviction grew that the problem of the continuum was in principle unsolvable. It was only after a way had been found of reducing mathematical concepts to set-theoretical ones, axioms had been stated in set-theoretical language which could be placed at the foundations of mathematical proofs actually encountered in real life and logical derivation methods had been formalized, that it became possible to give a precise statement, and then to solve the question, of the formal unsolvability of the continuum hypothesis. Formal unsolvability is understood in the sense that there does not exist a formal derivation in the Zermelo–Fraenkel system ZF either for the continuum hypothesis or for its negation.

In 1939 K. Gödel established the unprovability of the negation of the generalized continuum hypothesis (and hence the unprovability of the negation of the continuum hypothesis) in the system ZF with the axiom of choice (the system ZFC) under the hypothesis that ZF is consistent (see Gödel constructive set ). In 1963 P. Cohen showed that the continuum hypothesis (and therefore also the generalized continuum hypothesis) cannot be deduced from the axioms of ZFC assuming the consistency of ZF (see Forcing method ).

Are these results concerning the problem of the continuum final? The answer to this question depends on one's relation to the premise concerning the consistency of ZF and, what is more significant, to the experimental fact that every meaningful mathematical proof (of traditional classical mathematics) can, after it has been found, be adequately stated in the system ZFC. This fact cannot be proved nor can it even be precisely stated, since each revision raises a similar question concerning the adequacy of the revision for the revised theorem.

In model-theoretic language, Gödel and Cohen constructed models for ZFC in which $$ 2^{\mathfrak{k}} = \begin{cases} \mathfrak{m} & \text{if}\ \mathfrak{k} < \mathfrak{m}\,; \\ \mathfrak{k}^{+} & \text{if}\ \mathfrak{k} \ge \mathfrak{m} \ . \end{cases} $$

where $\mathfrak{m}$ is an arbitrary uncountable regular cardinal number given in advance, and $\mathfrak{k}^{+}$ is the first cardinal number greater than $\mathfrak{k}$. What is the possible behaviour of the function $2^{\mathfrak{k}}$ in various models of ZFC?

It is known that for regular cardinal numbers $\mathfrak{k}$, this function can take them to arbitrary cardinal numbers subject only to the conditions $$ \mathfrak{k} < \mathfrak{k}' \Rightarrow 2^{\mathfrak{k}} < 2^{\mathfrak{k}'} \,,\ \ \ \mathfrak{k} < \text{cf}(\mathfrak{k}) $$ where $\text{cf}(\mathfrak{a})$ is the smallest cardinal number cofinal with $\mathfrak{a}$ (see Cardinal number ). For singular (that is, non-regular) $\mathfrak{k}$, the value of the function $2^{\mathfrak{k}}$ may depend on its behaviour at smaller cardinal numbers. E.g., if \eqref{eq:1} holds for all $\alpha < \omega_1$, then it also holds for $\alpha = \omega_1$.

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Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH How far wrong could the Continuum Hypothesis be? When was the continuum hypothesis born?

The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ \aleph_1=2^{\aleph_0}$ . In other words, it asserts that every subset of the set of real numbers that contains the natural numbers has either the cardinality of the natural numbers or the cardinality of the real numbers. It was the first problem on the 1900 Hilbert's list of problems. The generalized continuum hypothesis asserts that there are no intermediate cardinals between every infinite set X and its power set.

Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

The question

My question asks for a description and explanation of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers ( part 1 , part 2 ).

Another by Shelah (perhaps in this paper entitled "The Generalized Continuum Hypothesis revisited " ). See also the paper entitled " You can enter Cantor paradise " (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's answer is by Lucca Belloti (also suggested by Haim ).

The proposed answer $ 2^{\aleph_0}=\aleph_2$ goes back according to François to Goedel . It is (perhaps) mentioned in Foreman's presentation. (I heard also from Menachem Magidor that this answer might have some advantages.)

François G. Dorais mentioned an important paper by Todorcevic's entitled " Comparing the Continuum with the First Two Uncountable Cardinals ".

There is also a very rich theory (PCF theory) of cardinal arithmetic which deals with what can be proved in ZFC.

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

There are by now a couple of long detailed excellent answers (that I still have to digest) by Joel David Hamkins and by Andres Caicedo and several other useful answers. (Unfortunately, I can accept only one answer.)

Update (February 2011): A new detailed answer was contributed by Justin Moore .

Update (Oct 2013) A user 'none' gave a link to an article by Peter Koellner about the current status of CH :

Update (Jan 2014) A related popular article in "Quanta:" To settle infinity dispute a new law of logic

(belated) update (Jan 2014) Joel David Hamkins links in a comment from 2012 a very interesting paper Is the dream solution to the continuum hypothesis attainable written by him about the possibility of a "dream solution to CH." A link to the paper and a short post can be found here .

(belated) update (Sept 2015) Here is a link to an interesting article: Can the Continuum Hypothesis be Solved? By Juliette Kennedy

Update A videotaped lecture The Continuum Hypothesis and the search for Mathematical Infinity by Woodin from January 2015, with reference also to his changed opinion. (added May 2017)

Update (Dec '15): A very nice answer was added (but unfortunately deleted by owner, (2019) now replaced by a new answer) by Grigor. Let me quote its beginning (hopefully it will come back to life):

"One probably should add that the continuum hypothesis depends a lot on how you ask it.

• $2^{\omega}=\omega_1$
• Every set of reals is either countable or has the same size as the continuum.

To me, 1 is a completely meaningless question, how do you even experiment it?

If I am not mistaken, Cantor actually asked 2..."

Update A 2011 videotaped lecture by Menachem Magidor: Can the Continuum Problem be Solved? (I will try to add slides for more recent versions.)

Update (July 2019) Here are slides of 2019 Woodin's lecture explaining his current view on the problem. (See the answer of Mohammad Golshani.)

Update (Sept 19, 2019) Here are videos of the three 2016 Bernay's lectures by Hugh Woodin on the continuum hypothesis and also the videos of the three 2012 Bernay's lectures on the continuum hypothesis and related topics by Solomon Feferman .

Update (Sept '20) Here are videos of the three 2020 Bernays' lectures by Saharon Shelah on the continuum hypothesis.

Update (May '21) In a new answer , Ralf Schindler gave a link to his 2021 videotaped lecture in Wuhan, describing a result with David Asperó that shows a relation between two well-known axioms. It turns out that Martin's Maximum $^{++}$ implies Woodin's ℙ $_{max}$ axiom. Both these axioms were known to imply the $\aleph_2$ answer to CH. A link to the paper: https://doi.org/10.4007/annals.2021.193.3.3

• continuum-hypothesis

• 9 $\begingroup$ Another noteworthy reference is Gödel's unpublished note 1970a where he collects evidence in favor of $2^{\aleph_0} = \aleph_2$ - books.google.com/… $\endgroup$ –  François G. Dorais Commented May 7, 2010 at 8:00
• 3 $\begingroup$ Another important paper is Todorcevic's Comparing the Continuuum with the First Two Uncountable Cardinals - math.toronto.edu/stevo $\endgroup$ –  François G. Dorais Commented May 7, 2010 at 8:02
• 4 $\begingroup$ Can you please edit your second sentence starting by "In other words" ? "containing of real numbers" does not seem very clear. Also small typos like "now intermediate" -> "no intermediate" "Eralier"=>"Earlier" Another pedantic remark is that one does not "solve" an hypothesis. One adopts it or one rejects it, eventually replacing it by another. Perhaps should you slightly rephrase your question. $\endgroup$ –  ogerard Commented May 8, 2010 at 12:40
• 10 $\begingroup$ Gil, you're absolutely right that using "solution" in this and related contexts is common usage, but IMHO it's bad English. One solves a problem, answers a question, and proves or disproves a hypothesis or conjecture. Whenever I hear that a "conjecture" has been "solved" I don't know whether the speaker means the conjecture is now known to be true, false, or undecidable (and I've seen examples in which each of the three was meant). But I'm grumpier about such things than most. $\endgroup$ –  Mark Meckes Commented May 10, 2010 at 13:36
• 4 $\begingroup$ There is an honorable precedent for calling it Cantor's Continuum Problem . $\endgroup$ –  bof Commented Jan 3, 2014 at 12:43

12 Answers 12

Since you have already linked to some of the contemporary primary sources, where of course the full accounts of those views can be found, let me interpret your question as a request for summary accounts of the various views on CH. I'll just describe in a few sentences each of what I find to be the main issues surrounding CH, beginning with some historical views. Please forgive the necessary simplifications.

Cantor. Cantor introduced the Continuum Hypothesis when he discovered the transfinite numbers and proved that the reals are uncountable. It was quite natural to inquire whether the continuum was the same as the first uncountable cardinal. He became obsessed with this question, working on it from various angles and sometimes switching opinion as to the likely outcome. Giving birth to the field of descriptive set theory, he settled the CH question for closed sets of reals, by proving (the Cantor-Bendixon theorem) that every closed set is the union of a countable set and a perfect set. Sets with this perfect set property cannot be counterexamples to CH, and Cantor hoped to extend this method to additional larger classes of sets.

Hilbert. Hilbert thought the CH question so important that he listed it as the first on his famous list of problems at the opening of the 20th century.

Goedel. Goedel proved that CH holds in the constructible universe $L$ , and so is relatively consistent with ZFC. Goedel viewed $L$ as a device for establishing consistency, rather than as a description of our (Platonic) mathematical world, and so he did not take this result to settle CH. He hoped that the emerging large cardinal concepts, such as measurable cardinals, would settle the CH question, and as you mentioned, favored a solution of the form $2^\omega=\aleph_2$ .

Cohen. Cohen introduced the method of forcing and used it to prove that $\neg$ CH is relatively consistent with ZFC. Every model of ZFC has a forcing extension with $\neg$ CH. Thus, the CH question is independent of ZFC, neither provable nor refutable. Solovay observed that CH also is forceable over any model of ZFC.

Large cardinals. Goedel's expectation that large cardinals might settle CH was decisively refuted by the Levy-Solovay theorem, which showed that one can force either CH or $\neg$ CH while preserving all known large cardinals. Thus, there can be no direct implication from large cardinals to either CH or $\neg$ CH. At the same time, Solovay extended Cantor's original strategy by proving that if there are large cardinals, then increasing levels of the projective hierarchy have the perfect set property, and therefore do not admit counterexamples to CH. All of the strongest large cardinal axioms considered today imply that there are no projective counterexamples to CH. This can be seen as a complete affirmation of Cantor's original strategy.

Basic Platonic position. This is the realist view that there is Platonic universe of sets that our axioms are attempting to describe, in which every set-theoretic question such as CH has a truth value. In my experience, this is the most common or orthodox view in the set-theoretic community. Several of the later more subtle views rest solidly upon the idea that there is a fact of the matter to be determined.

Old-school dream solution of CH. The hope was that we might settle CH by finding a new set-theoretic principle that we all agreed was obviously true for the intended interpretation of sets (in the way that many find AC to be obviously true, for example) and which also settled the CH question. Then, we would extend ZFC to include this new principle and thereby have an answer to CH. Unfortunately, no such conclusive principles were found, although there have been some proposals in this vein, such as Freilings axiom of symmetry .

Formalist view. Rarely held by mathematicians, although occasionally held by philosophers, this is the anti-realist view that there is no truth of the matter of CH, and that mathematics consists of (perhaps meaningless) manipulations of strings of symbols in a formal system. The formalist view can be taken to hold that the independence result itself settles CH, since CH is neither provable nor refutable in ZFC. One can have either CH or $\neg$ CH as axioms and form the new formal systems ZFC+CH or ZFC+ $\neg$ CH. This view is often mocked in straw-man form, suggesting that the formalist can have no preference for CH or $\neg$ CH, but philosophers defend more subtle versions, where there can be reason to prefer one formal system to another.

Pragmatic view. This is the view one finds in practice, where mathematicians do not take a position on CH, but feel free to use CH or $\neg$ CH if it helps their argument, keeping careful track of where it is used. Usually, when either CH or $\neg$ CH is used, then one naturally inquires about the situation under the alternative hypothesis, and this leads to numerous consistency or independence results.

Cardinal invariants. Exemplifying the pragmatic view, this is a very rich subject studying various cardinal characteristics of the continuum, such as the size of the smallest unbounded family of functions $f:\omega\to\omega$ , the additivity of the ideal of measure-zero sets, or the smallest size family of functions $f:\omega\to\omega$ that dominate all other such functions. Since these characteristics are all uncountable and at most the continuum, the entire theory trivializes under CH, but under $\neg$ CH is a rich, fascinating subject.

Canonical Inner models. The paradigmatic canonical inner model is Goedel's constructible universe $L$ , which satisfies CH and indeed, the Generalized Continuum Hypothesis, as well as many other regularity properties. Larger but still canonical inner models have been built by Silver, Jensen, Mitchell, Steel and others that share the GCH and these regularity properties, while also satisfying larger large cardinal axioms than are possible in $L$ . Most set-theorists do not view these inner models as likely to be the "real" universe, for similar reasons that they reject $V=L$ , but as the models accommodate larger and larger large cardinals, it becomes increasingly difficult to make this case. Even $V=L$ is compatible with the existence of transitive set models of the very largest large cardinals (since the assertion that such sets exist is $\Sigma^1_2$ and hence absolute to $L$ ). In this sense, the canonical inner models are fundamentally compatible with whatever kind of set theory we are imagining.

Woodin. In contrast to the Old-School Dream Solution, Woodin has advanced a more technical argument in favor of $\neg$ CH. The main concepts include $\Omega$ -logic and the $\Omega$ -conjecture, concerning the limits of forcing-invariant assertions, particularly those expressible in the structure $H_{\omega_2}$ , where CH is expressible. Woodin's is a decidedly Platonist position, but from what I have seen, he has remained guarded in his presentations, describing the argument as a proposal or possible solution, despite the fact that others sometimes characterize his position as more definitive.

Foreman. Foreman, who also comes from a strong Platonist position, argues against Woodin's view. He writes supremely well, and I recommend following the links to his articles.

Multiverse view. This is the view, offered in opposition to the Basic Platonist Position above, that we do not have just one concept of set leading to a unique set-theoretic universe, but rather a complex variety of set concepts leading to many different set-theoretic worlds. Indeed, the view is that much of set-theoretic research in the past half-century has been about constructing these various alternative worlds. Many of the alternative set concepts, such as those arising by forcing or by large cardinal embeddings are closely enough related to each other that they can be compared from the perspective of each other. The multiverse view of CH is that the CH question is largely settled by the fact that we know precisely how to build CH or $\neg$ CH worlds close to any given set-theoretic universe---the CH and $\neg$ CH worlds are in a sense dense among the set-theoretic universes. The multiverse view is realist as opposed to formalist, since it affirms the real nature of the set-theoretic worlds to which the various set concepts give rise. On the Multiverse view, the Old-School Dream Solution is impossible, since our experience in the CH and $\neg$ CH worlds will prevent us from accepting any principle $\Phi$ that settles CH as "obviously true". Rather, on the multiverse view we are to study all the possible set-theoretic worlds and especially how they relate to each other.

I should stop now, and I apologize for the length of this answer.

• 16 $\begingroup$ it seems that the multiverse view is the beginning of plurality in set theory. This is analogous to how there used to be only one geometry -- Euclidean -- but the investigation of PP and not PP led to a multiverse of geometries. $\endgroup$ –  user2529 Commented May 19, 2010 at 8:32
• 8 $\begingroup$ I agree, Colin; the analogy with geometry is very strong and extends to many facets of how we think about the various geometries. $\endgroup$ –  Joel David Hamkins Commented May 19, 2010 at 16:18
• 36 $\begingroup$ "Formalist view. Rarely held by mathematicians, although occasionally held by philosophers ..." I recently read the following, written by a distinguished philosopher of mathematics. "The formalist's central thought is that arithmetic is not ultimately concerned with an extralinguistic domain of things. Rather, insofar as arithmetic has a proper subject matter, it is the language of arithmetic itself and certain formal relations among its sentences." This was accompanied by a footnote: "The view has few contemporary adherents among philosophers, though mathematicians often find it congenial." $\endgroup$ –  gowers Commented Feb 6, 2011 at 17:28
• 17 $\begingroup$ For what it's worth, I find it congenial myself. In particular, I incline to the view that there is no fact of the matter about whether CH is true. $\endgroup$ –  gowers Commented Feb 6, 2011 at 17:29
• 13 $\begingroup$ "Formalist view. Rarely held by mathematicians"? Do you have any evidence for your claim? I would've said formalism was the prevalent view among mathematicians (exept for the adherents to that strange religion called "platonism"), but I have no evidence to substantiate this: mine is probably just wishful thinking... $\endgroup$ –  Qfwfq Commented Dec 16, 2015 at 20:50

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain there.

You can find the slides here , under "Recent results about the Continuum Hypothesis, after Woodin". (In true Bourbaki fashion, I heard that the talk was not well received.)

Roughly, Woodin's approach shows that in a sense, the theory of $H(\omega_2)$ decided by the usual set of axioms, ZFC and large cardinals, can be "finitely completed" in a way that would make it reasonable to expect to settle all its properties. However, any such completion implies the negation of CH.

It is a conditional result, depending on a highly non-trivial problem, the $\Omega$-conjecture. If true, this conjecture gives us that Cohen's technique of forcing is in a sense the only method (in the presence of large cardinals) required to establish consistency. (The precise statement is more technical.)

$H(\omega_2)$, that Dehornoy calls $H_2$, is the structure obtained by considering only those sets $X$ such that $X\cup\bigcup X\cup\bigcup\bigcup X\cup\dots$ has size strictly less than $\aleph_2$, the second uncountable cardinal.

Replacing $\aleph_2$ with $\aleph_1$, we have $H(\omega_1)$, whose theory is completely settled in a sense, in the presence of large cardinals. If nothing else, one can think of Woodin's approach as trying to build an analogy with this situation, but "one level up."

Whether or not one considers that settling the $\Omega$-conjecture in a positive fashion actually refutes CH in some sense, is a delicate matter. In any case (and I was happy to see that Dehornoy emphasizes this), Woodin's approach gives strength to the position that the question of CH is meaningful (as opposed to simply saying that, since it is independent, there is nothing to decide).

(2) There is another approach to the problem, also pioneered by Hugh Woodin. It is the matter of "conditional absoluteness." CH is a $\Sigma^2_1$ statement. Roughly, this means that it has the form: "There is a set of reals such that $\phi$", where $\phi$ can be described by quantifying only over reals and natural numbers. In the presence of large cardinals, Woodin proved the following remarkable property: If $A$ is a $\Sigma^2_1$ statement, and we can force $A$, then $A$ holds in any model of CH obtained by forcing.

Recall that forcing is essentially the only tool we have to establish consistency of statements. Also, there is a "trivial" forcing that does not do anything, so the result is essentially saying that any statement of the same complexity as CH, if it is consistent (with large cardinals), then it is actually a consequence of CH.

This would seem a highly desirable ''maximality'' property that would make CH a good candidate to be adopted.

However, recent results (by Aspero, Larson, and Moore) suggest that $\Sigma^2_1$ is close to being the highest complexity for which a result of this kind holds, which perhaps weakens the argument for CH that one could do based on Hugh's result.

A good presentation of this theorem is available in Larson's book "The stationary tower. Notes on a Course by W. Hugh Woodin." Unfortunately, the book is technical.

(3) Foreman's approach is perhaps the strongest opponent to the approach suggested by Woodin in (1) . Again, it is based in the technique of forcing, now looking at small cardinal analogues of large cardinal properties.

Many large cardinal properties are expressed in terms of the existence of elementary embeddings of the universe of sets. These embeddings tend to be "based" at cardinals much much larger than the size of the reals. With forcing, one can produce such embeddings "based" at the size of the reals, or nearby. Analyzing a large class of such forcing notions, Foreman shows that they must imply CH. If one were to adopt the consequences of performing these forcing constructions as additional axioms one would then be required to also adopt CH.

I had to cut my answer short last time. I would like now to say a few details about a particular approach.

(4) Forcing axioms imply that $2^{\aleph_0}=\aleph_2$, and (it may be argued) strongly suggest that this should be the right answer.

Now, before I add anything, note that Woodin's approach (1) uses forcing axioms to prove that there are "finite completions" of the theory of $H(\omega_2)$ (and the reals have $\aleph_2$). However, this does not mean that all such completions would be compatible in any particular sense, or that all would decide the size of the reals. What Woodin proves is that all completions negate CH, and forcing axioms show that there is at least one such completion.

I believe there has been some explanation of forcing axioms in the answer to the related question on GCH. Briefly, the intuition is this: ZFC seems to capture the basic properties of the universe of sets, but fails to account for its width and its height . (What one means by this is: how big should power sets be, and how many ordinals there are.)

Our current understanding suggests that the universe should indeed be very tall, meaning there should be many many large cardinals. As Joel indicated, there was originally some hope that large cardinals would determine the size of the reals, but just about immediately after forcing was introduced, it was proved that this was not the case. (Technically, small forcing preserves large cardinals.)

However, large cardinals settle many questions about the structure of the reals (all first order, or projective statements, in fact). CH, however, is "just" beyond what large cardinals can settle. One could say that, as far as large cardinals are concerned, CH is true. What I mean is that, in the presence of large cardinals, any definable set of reals (for any reasonable notion of definability) is either countable or contains a perfect subset. However, this may simply mean that there is certain intrinsic non-canonicity in the sets of reals that would disprove CH, if this is the case.

(A word of caution is in order here, and there are candidates for large cardinal axioms [presented by Hugh Woodin in his work on suitable extender sequences] for which preservation under small forcing is not clear. Perhaps the solution to CH will actually come, unexpectedly, from studying these cardinals. But this is too speculative at the moment.)

I have avoided above saying much about forcing. It is a massive machinery, and any short description is bound to be very inaccurate, so I'll be more than brief.

An ordinary algebraic structure (a group, he universe of sets) can be seen as a bi-valued model. Just as well, one can define, for any complete Boolean algebra ${\mathbb B}$, the notion of a structure being ${\mathbb B}$-vaued. If you wish, "fuzzy set theory" is an approximation to this, as are many of the ways we model the world by using a probability density to decide the likelihood of events. For any complete Boolean algebra ${\mathbb B}$, we can define a ${\mathbb B}$-valued model $V^{\mathbb B}$ of set theory. In it, rather than having for sets $x$ and $y$ that either $x\in y$ or it doesn't, we assign to the statement $x\in y$ a value $[x\in y]\in{\mathbb B}$. The way the construction is performed, $[\phi]=1$ for each axiom $\phi$ of ZFC. Also, for each element $x$ of the actual universe of sets, there is a copy $\check x$ in the ${\mathbb B}$-valued model, so that the universe $V$ is a submodel of $V^{\mathbb B}$. If it happens that for some statement $\psi$ we have $[\psi]>0$, we have established that $\psi$ is consistent with ZFC. By carefully choosing ${\mathbb B}$, we can do this for many $\psi$. This is the technique of forcing, and one can add many wrinkles to the approach just outlined. One refers to ${\mathbb B}$ as a forcing notion .

Now, the intuition that the universe should be very fat is harder to capture than the idea of largeness of the ordinals. One way of expressing it is that the universe is somehow "saturated": If the existence of some object is consistent in some sense, then in fact such object should exist. Formalizing this, one is led to forcing axioms. A typical forcing axiom says that relatively simple (under some measure of complexity) statements that can be shown consistent using the technique of forcing via a Boolean algebra ${\mathbb B}$ that is not too pathological, should in fact hold.

The seminal Martin's Maximum paper of Foreman-Magidor-Shelah identified the most generous notion of "not too pathological", it corresponds to the class of "stationary set preserving" forcing notions. The corresponding forcing axiom is Martin's Maximum, MM. In that paper, it was shown that MM implies that the size of the reals is $\aleph_2$.

The hypothesis of MM has been significantly weakened, through a series of results by different people, culminating in the Mapping Reflection Principle paper of Justin Moore. Besides this line of work, many natural consequences of forcing axioms (commonly termed reflection principles ) have been identified, and shown to be independent of one another. Remarkably, just about all these principles either imply that the size of the reals is $\aleph_2$, or give $\aleph_2$ as an upper bound.

Even if one finds that forcing axioms are too blunt a way of capturing the intuition of "the universe is wide", many of its consequences are considered very natural. (For example, the singular cardinal hypothesis , but this is another story.) Just as most of the set theoretic community now understands that large cardinals are part of what we accept about the universe of sets (and therefore, so is determinacy of reasonably definable sets of reals, and its consequences such us the perfect set property), it is perhaps not completely off the mark to expect that as our understanding of reflection principles grow, we will adopt them (or a reasonable variant) as the right way of formulating "wideness". Once/if that happens, the size of the reals will be taken as $\aleph_2$ and therefore CH will be settled as false.

The point here is that this would be a solution to the problem of CH that does not attack CH directly. Rather, it turns out that the negation of CH is a common consequence of many principles that it may be reasonable to adapt in light of the naturalness of some of their best known consequences, and of their intrinsic motivation coming from the "wide universe" picture.

(Apologies for the long post.)

Edit, Nov. 22/10: I have recently learned about Woodin's "Ultimate L" which, essentially, advances a view that theories are "equally good" if they are mutually interpretable , and identifies a theory ("ultimate L") that, modulo large cardinals, would work as a universal theory from which to interpret all extensions. This theory postulates an $L$-like structure for the universe and in particular implies CH, see this answer . But, again, the theory is not advocated on grounds that it ought to be true , whatever this means, but rather, that it is "richest" possible in that it allows us to interpret all possible "natural" extensions of ZFC. In particular, under this approach, only large cardinals are relevant if we want to strengthen the theory, while "width" considerations, such as those supporting forcing axioms, are no longer relevant.

Since the approach I numbered (1) above implies the negation of CH, I feel I should add that one of the main reasons for it being advanced originally depended on the fact that the set of $\Omega$-validities can be defined "locally", at the level of $H({\mathfrak c}^+)$, at least if the $\Omega$-conjecture holds.

However, recent results of Sargsyan uncovered a mistake in the argument giving this local definability. From what I understand, Woodin feels that this weakens the case he was making for not-CH significantly.

Added link: slides of a 2016 lecture by Woodin on Ultimate L.

• 2 $\begingroup$ Thanks very much for this answer, Andres. I had heard that Woodin proved an extremal property of CH, but I didn't know what it was. It is presumably your item (2). $\endgroup$ –  John Stillwell Commented May 18, 2010 at 23:47
• 2 $\begingroup$ Hi Simon. I am not too certain of all the details; I believe that the issue is this: In the context of $AD^+$, say that $\alpha$ is a "local $\Theta$" if it is the $\Theta$ of a hod-model. Woodin had an argument showing that no such $\alpha$ could be overlapped by a strong cardinal. This put serious limitations on the strength of large cardinals that hod-models could contain. In particular, this was the reason why it was expected that "CH + there is an $\omega_1$-dense ideal on $\omega_1$" and "$AD_{\mathbb R}+\Theta$ regular" were expected to have really high consistency strength (continued) $\endgroup$ –  Andrés E. Caicedo Commented Nov 23, 2010 at 15:33
• 2 $\begingroup$ (2) Woodin's local definability argument depended on this limitation of hod models. (I am not sure of the details here.) Grigor's analysis in the context of the core model induction (there are slides of a talk at Boise on his website, and I can email you his thesis, let me know) showed that these "local overlaps" are possible, and deduced as a corollary that "$AD_{\mathbb R}+\Theta$ regular has much lower consistency strength than expected. Grigor in fact has a very detailed analysis of hod models. Without the overlap limitation, the set of $\Omega$-validities ends up being harder to define. $\endgroup$ –  Andrés E. Caicedo Commented Nov 23, 2010 at 15:42
• 2 $\begingroup$ (3) It can be shown that it is $H(\delta_0^+)$-definable, where $\delta_0$ is the smallest Woodin of $V$. But the argument pro-not-CH went by a level by level analysis of the $H(\kappa)$-levels, and this jump (from ${\mathfrak c}$ to a Woodin) is too high to overlook. As far as I understand, this is the nature of the issue. $\endgroup$ –  Andrés E. Caicedo Commented Nov 23, 2010 at 15:45
• 1 $\begingroup$ @Joel Thanks. I am aware of this (see here ), and I knew there was a post of mine where I was using the incorrect terminology, but could never find it. :-) I'll update a bit later. $\endgroup$ –  Andrés E. Caicedo Commented Mar 27, 2018 at 19:46

First I'll say a few words about forcing axioms and then I'll answer your question. Forcing axioms were developed to provide a unified framework to establish the consistency of a number of combinatorial statements, particularly about the first uncountable cardinal. They began with Solovay and Tennenbaum's proof of the consistency of Souslin's Hypothesis (that every linear order in which there are no uncountable families of pairwise disjoint intervals is necessarily separable). Many of the consequences of forcing axioms, particularly the stronger Proper Forcing Axiom and Martin's Maximum, had the form of classification results: Baumgartner's classification of the isomorphism types of $\aleph_1$ -dense sets of reals, Abraham and Shelah's classification of the Aronszajn trees up to club isomorphism, Todorcevic's classification of linear gaps in $\mathcal{P}(\mathbb{N})/\mathrm{fin}$ , and Todorcevic's classification of transitive relations on $\omega_1$ . A survey of these results (plus many references) can be found in both Stevo Todorcevic's ICM article and my own (the later can be found here 1 ). These are accessible to a general audience.

What does all this have to do with the Continuum Problem? It was noticed early on that forcing axioms imply that the continuum is $\aleph_2$ . The first proof of this is, I believe, in Foreman, Magidor, and Shelah's seminal paper in Martin's Maximum. Other quite different proofs were given by Caicedo, Todorcevic, Velickovic, and myself. An elementary proof which is purely Ramsey-theoretic in nature is in my article " Open colorings, the continuum, and the second uncountable cardinal " (PAMS, 2002). 2

Since it is often the case that the combinatorial essence of the proofs of these classification results and that the continuum is $\aleph_2$ are similar, one is left to speculate that perhaps there may some day be a classification result concerning structures of cardinality $\aleph_1$ which already implies that the continuum is $\aleph_2$ . There is a candidate for such a classification: the assertion that there are five uncountable linear orders such that every other contains an isomorphic copy of one of these five. Another related candidate for such a classification is the assertion that the Aronszajn lines are well quasi-ordered by embeddability (if $L_i$ $(i < \infty)$ is a sequence of Aronszajn lines, then there is an $i < j$ such that $L_i$ embeds into $L_j$ ). These are due to myself and Carlos Martinez, respectively. See a discussion of this (with references) in my ICM paper 1 .

1 The Proper Forcing Axiom, Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010, Wayback Machine 2 Open colorings, the continuum and the second uncountable cardinal, Wayback Machine

Regarding Shelah's approach, I believe that the following paper should be quite accessible to non-professionals: YOU CAN ENTER CANTOR’S PARADISE!

Now, I have no idea how to explain Woodin's approach to CH without relying on some cryptic terminology, but I believe that the following paper by Luca Bellotti might be useful: Woodin on the Continuum Problem: an overview and some objections .

• $\begingroup$ Thanks Haim, I think I vaguely had this paper of Shelah also in mind but did not find it when I wrote the question. $\endgroup$ –  Gil Kalai Commented May 8, 2010 at 8:54
• 1 $\begingroup$ The second link is now broken. (I added the nice title for the first link.) $\endgroup$ –  Gil Kalai Commented Apr 30, 2019 at 19:09
• 1 $\begingroup$ @Gil I fixed the link. $\endgroup$ –  Andrés E. Caicedo Commented Apr 30, 2019 at 22:26

The question of whether or not $2^{\aleph_{0}} = \aleph_{1}$ is not even considered in Shelah's approach. In fact, this question is regarded as part of the "white noise" which has distracted the attention of set theorists from some striking $ZFC$-results about cardinal exponentiation $\kappa^{\lambda}$ when you consider relatively small exponents $\lambda$ and relatively large bases $\kappa$.

Let me add in short details, views of three famous set theorists about the problem:

Shelah’s answer : The question was wrong. The right question should be about other combinatorial objects. There we can prove the “revised GCH” (Sh460). PCF Theory .

Foreman’s answer : Large cardinals can’t help, but “generic large cardinals” might.

Woodin’s answer : Instead of looking at the statements of new axioms, look at the metamathematical properties of axiom candidates. There is an asymmetry between axioms that imply CH and those that imply $\sim CH.$ Woodin’s ­ $\Omega$ -conjecture .

Edit As it is stated in some other answers, Woodin has changed his mind and he believes in the continuum hypothesis. This is part of his "Ultimaate L" project. See the following slide for his very recent expository talk on the $CH$ :

The Continuum hypothesis

• $\begingroup$ Mohammad Golshani kindly mentioned in a FB discussion two recent approaches: "Matteo Viale has arguments and claims the continuum is $\aleph_2$. Sakae Fuchino claims the continuum is either $\aleph_1$ or $\aleph_2$ or is very large. $\endgroup$ –  Gil Kalai Commented Sep 11, 2020 at 6:34
• $\begingroup$ (cont.) For Viale see his papers Tameness for set theory I and Tameness for set theory II or the slide logicatorino.altervista.org/.../TAMSTslides.pdf $\endgroup$ –  Gil Kalai Commented Sep 11, 2020 at 6:35
• $\begingroup$ (cont.) For Fuchino's work see fuchino.ddo.jp/slides/kobe2020-07-28-pf.pdf $\endgroup$ –  Gil Kalai Commented Sep 11, 2020 at 6:36

In Aug. 2020, I gave a talk at Wuhan with the title "How many real numbers are there?", taking into account my result with D. Asperó on MM++ => (*). There is a recording: https://m.bilibili.com/video/BV1TV411h714 , and there is also a set of notes: https://ivv5hpp.uni-muenster.de/u/rds/Notes_Wuhan.pdf . Comments are very welcome.

• $\begingroup$ To answer the question in the title of your talk: 4, obviously. $\endgroup$ –  Asaf Karagila ♦ Commented May 3, 2021 at 10:29
• 1 $\begingroup$ The paper in question was just published in the Annals: doi.org/10.4007/annals.2021.193.3.3 $\endgroup$ –  Sam Hopkins Commented May 3, 2021 at 15:47
• 2 $\begingroup$ A quanta magazine article: quantamagazine.org/… $\endgroup$ –  Gil Kalai Commented Jul 16, 2021 at 11:32
• 1 $\begingroup$ Hi Ralf, I think the problem is your bias, no? When one works under one theory the other looks unnatural, when one switches back and fourth between both MM and Ultimate-L then both seem natural. Its a choice people make, they choose not to get into the other theory, but surely we cannot be biased this way towards the cardinality of R. More objective arguments are needed. $\endgroup$ –  Grigor Commented Jan 12, 2022 at 23:36
• 1 $\begingroup$ For example, one can now ask what is $2^{\omega_2}$? If $2^\omega=\omega_2$ and if we know this then surely we must also have a definite answer for $2^{\omega_2}$ and $2^{2^{\omega_2}}$ and etc . MM is silent on those. $\endgroup$ –  Grigor Commented Jan 12, 2022 at 23:41

This is a nice survey:

• http://logic.harvard.edu/EFI_CH.pdf

Koellner, Peter (2011). "The Continuum Hypothesis". Exploring the Frontiers of Independence (Harvard lecture series).

(Pasted from Wikipedia article on CH.)

Since this old question has been bumped to the front page, I'm surprised nobody so far mentioned the following opinion expressed by Paul Cohen at the end of his 1966 book Set Theory and the Continuum Hypothesis :

A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it is absurd to think that the process of adding one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now $\aleph_1$ is the set of countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set $\mathfrak{c}$ is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach $\mathfrak{c}$. Thus $\mathfrak{c}$ is greater than $\aleph_n$, $\aleph_\omega$, $\aleph_\alpha$ where $\alpha = \aleph_\omega$ etc. This point of view regards $\mathfrak{c}$ as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.

(The emphasis is Cohen's. The $\mathfrak{c}$ is actually a plain C in the typewritten text.)

• 1 $\begingroup$ A similar view is embraced by Dana Scott in the foreword to J. L. Bell's Boolean-valued models and independence proofs in set theory , Clarendon Press, Oxford, 1977. He writes `` Perhaps we would be pushed in the end to say that all sets are countable (and that the continuum is not even a set) when at last all cardinals are absolutely destroyed.'' $\endgroup$ –  Andrés E. Caicedo Commented May 14, 2017 at 13:23
• $\begingroup$ FWIW, I'm not sure that Cohen held fast to that opinion throughout his life. Elsewhere he said, "Well, philosophically, if you really believe sets exist -- I mean, if you adopt the extreme platonic position -- you can ask, what is the answer? Certainly Goedel himself had the platonic view that the question demanded an absolute answer and that, therefore, neither his proof of the consistency of the continuum hypothesis nor mine of its independence from them was a final answer. My personal view is that I regard the present solution of the problem as very satisfactory. (cont.) $\endgroup$ –  Todd Trimble ♦ Commented May 14, 2017 at 14:47
• 1 $\begingroup$ "I think it is the only possible solution. It gives one a feeling for what's possible and what's impossible, and in that sense I feel one should be very satisfied... If I were a betting man, I'd bet no one is going to come up with any other kind of solution. There will be philosophical papers, but I don't think any mathematical paper will say that there is any answer other than the answer that it's undecidable." (Interview in July 1985 for More Mathematical People.) $\endgroup$ –  Todd Trimble ♦ Commented May 14, 2017 at 14:51

my answer was too fast, and so i deleted it, but it seems that Gil refers to it, so i will again very quickly explain the matter.

in general, i agree with the view that some mathematicians express, namely that there is no fact of the matter here. the issue has two aspects.

Aspect 1. Cantor originally did not ask an AC cardinality question, namely if $2^\omega=\omega_1$ . more or less at the beginning he was asking if the perfect set property is true. of course, then this becomes (under AC) a cardinality question.

the trouble with the AC question is that it seems as if one is asking how many real numbers there are, and this question, to me, makes no sense. i can only speculate, judging from Cantor's work elsewhere (Cantor-Bendixon analysis, closed sets have the perfect set property), that Cantor was originally concerned with properties that sets of reals have rather than "how many real numbers there are", where "how many" is asked the same way when one asks how many people there are in the room. to begin with, where exactly these real numbers are supposed to be? i simply cannot make sense of these sort of mysterious questions.

more down to earth question would be whether every set of reals has a perfect set property. here, the dividing line is perfect. Large Cardinals imply that every definable set has the perfect set property and the Axiom of Choice does what it always does, namely it produces a set, using a magical enumeration of the real numbers, without a perfect set property.

the question whether every "definable" set has the perfect set property is a mathematical question provided one fixes a notion of definability. the question is resolved under Large Cardinals for almost all reasonable notions of definability, projective, in $L(\mathbb{R})$ and etc.

in fact, the story is even better. Large Cardinals give exact answers to these questions. meaning, if you want all Projective Sets to behave the way Borel and Analytic sets behave (Lebesgue measurability, perfect set property, Baire property, determinacy and etc) then you must have these or that large cardinal in your universe hiding somewhere (usually in the form of a countable model that "captures" the level of definability in question).

In the context of definability, CH (even the cardinality version) is a mathematical rather than a philosophical question.

Aspect 2. if i may say so, usually two notions get mixed up. one is "the platonistic universe of sets" and the other is "foundations of mathematics".

the universe of sets is the way it is, it doesn't depend on what i or anybody else likes or doesn't like. the actual universe is also the way it is, it also doesn't depend on what i or anybody else likes, it is just the way it is.

it must be abundantly clear to anyone following physics that there are several ways of talking about the universe, the exact same universe that doesn't depend on any one person.

the parallel is exactly that. "foundations of mathematics" is the language we pick, according to our licking, to discuss the properties of the universe that we see or intuit.

my goal is to avoid debate here, so i will not mention specific views (except one mentioned just as a clarification), but most views expressed here are proposals for a choice of language, a way of speaking about sets. one view, which i will mention but not for the purpose of debating, the Multiverse View, is actually a mixture of language and the set theoretic reality which in my personal view is rather confused (i am not saying this to start a debate, forgive me for the word).

when interpreting MV as a language, namely as a type of foundations of math, one faces challenges that are outlined in Steel's "Goedel's Program". what is the actual logical language, what is the satisfaction relation and etc and etc, it is easy to declare success under MV but a lot is left unexplained.

when interpreting MV as a view of the universe, namely that the universe has that form, i am perplexed because no reason is given for knowing the existence of it. we only truly know the existence of one universe, namely Goedel's L. the rest, the forcing extensions, are nothing but generalized compactness constructions, the way we get non-standard numbers from the actual numbers. they exist in the language, we can talk about them, but asserting that they exist is going a long way. it is like saying that number theorists study all models of PA rather than the properties of (N, 0, 1, +, ., Suc, exp, <). i am not so sure if number theorists would agree with this.

at any rate, what exactly one is asking when one is asking whether CH is true or not?

one can ask if CH is true in the actual set theoretic universe independently of the language we use to talk about it, in some absolute sense. i believe this interpretation is not a meaningful interpretation unless CH or not CH is somehow implied (in the logical sense) by some basic principles of sets that one can discover, these principles must be independent of the language. i doubt this can happen. for all i know, ZFC may not even be true in the actual universe, it seems more plausible that the universe is expanding and every set is actually countable from some point of view, and so in the actual universe the set of reals may not exist.

so for me, CH, asked as a cardinality question, only makes sense when one is discussing language. does PFA imply it? does this or that foundational framework imply it? and etc.

the following point is perhaps overlooked by many set theorists and mathematicians.

the trouble for set theory isn't that there are undecidable questions. this just says something about our language being incomplete, which is not so difficult to accept. the trouble is that there could be two ways of speaking about reality with no way of interpreting one into another.

the parallel would be this. there could be creatures out there somewhere living in the same universe as we do, yet their laws of nature that make the exact same predictions as ours do, are simply not translatable into ours. meaning there is no way for us to understand what they are saying, yet we can see them do their physics which from our perspective is total gibberish that somehow makes the very same predictions as ours. this seems impossible, how would our laws explain their existence? it would seem that our laws are incomplete and we need to do more research to explain what they are saying.

it could be that PFA and Ultimate L line are examples of two ways of discussing the universe that are simply disconnected from one another. they do not discuss the same reality. this must be false, they both stem from large cardinals, they both are large cardinal theories in disguise. but how can we prove this or at least feel confident that they are the same.

i believe that Steel's Program is the most important program that reasonably deals with this issue. Steel's Program is an interpretation of Goedel's Program, and the idea is to develop tools to translate between mathematical frameworks. Steel views large cardinals and inner model theory in this sense, as a way of translating PFA into the realm of determinacy axioms and etc. the two big open problems in this line of research are the following.

• is there a natural determinacy theory T such that PFA can be forced over any "nice" model of T.
• does PFA imply that HOD of $L(\Gamma_{uB}, \mathbb{R})$ has superstrong cardinals.

i will not go any further than this.

There is a recent (and unfinished) attempt to view the Shelah's approach to Continuum Hypothesis in terms of homotopy theory.

• $\begingroup$ Please change a link, It does not work anymore. $\endgroup$ –  Evgeny Kuznetsov Commented Mar 20, 2017 at 6:19
• $\begingroup$ I've redirected the link to Gabrilovich's website, which has a number of relevant texts. $\endgroup$ –  Noah Schweber Commented Jun 16, 2017 at 2:23

Shelah's approach from his paper in his " The Generalized Continuum Hypothesis revisited ", concerns mainly the generalized continuum hypothesis. His main theorem addresses an appealing variation of the GCH which is based on a revised notion of "power." Let me explain what is this notion of $\lambda^{[\kappa]}$ (in words: $\lambda$ to the revised power of $\kappa$,) which is central to his approach and is also of independent interest. $\lambda^{[\kappa]}$ is the minimum size of a family of subsets of size $\kappa$ of a set $X$ of cardinality $\lambda$ such that every subset of cardinality $\kappa$ of $X$ is covered by less than $\kappa$ members of the family. (Of course we need that $\kappa < \lambda$ and also we need that $\kappa$ is a regular cardinal.)

The introduction to the paper is readable and motivated.

• $\begingroup$ The link in the post seems to be dead, here is a Wayback Machine snapshot from 2019. Some other links: shelah.logic.at/files/95804/460.pdf , arxiv.org/abs/math/9809200 , doi.org/10.1007/BF02773223 $\endgroup$ –  Martin Sleziak Commented Sep 1, 2021 at 5:50

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Can the Continuum Hypothesis Be Solved?

In 1900, David Hilbert published a list of twenty-three open questions in mathematics , ten of which he presented at the International Congress of Mathematics in Paris that year. Hilbert had a good nose for asking mathematical questions as the ones on his list went on to lead very interesting mathematical lives. Many have been solved, but some have not been, and seem to be quite difficult. In both cases, some very deep mathematics has been developed along the way. The so-called Riemann hypothesis, for example, has withstood the attack of generations of mathematicians ever since 1900 (or earlier). But the effort to solve it has led to some beautiful mathematics. Hilbert’s fifth problem turned out to assert something that couldn’t be true, though with fine tuning the “right” question—that is, the question Hilbert should have asked—was both formulated and solved. There is certainly an art to asking a good question in mathematics.

The problem known as the continuum hypothesis has had perhaps the strangest fate of all. The very first problem on the list, it is simple to state: how many points on a line are there? Strangely enough, this simple question turns out to be deeply intertwined with most of the interesting open problems in set theory , a field of mathematics with a very general focus, so general that all other mathematics can be seen as part of it, a kind of foundation on which the house of mathematics rests. Most objects in mathematics are infinite, and set theory is indeed just a theory of the infinite.

How ironic then that the continuum hypothesis is unsolvable—indeed, “provably unsolvable,” as we say. This means that none of the known mathematical methods—those that mathematicians actually use and find legitimate—will suffice to settle the continuum hypothesis one way or another. It seems odd that being unsolvable is the kind of thing one can prove about a mathematical question. In fact, there are many questions of this type, particularly about sets of real numbers—or sets of points on a line, if you like—that we know cannot be settled using standard mathematical methods.

Now, mathematics is not frozen in time or method—to the contrary, it is a very dynamic enterprise, each generation expanding and building on what went before. This process of expansion has not always been easy; sometimes it takes a while before new methods are accepted. This was true of set theory in the late nineteenth century. Its inventor, Georg Cantor , met with serious opposition on the part of those who were hesitant to admit infinite objects into mathematics.

What concerns us here is not so much the prehistory of the continuum hypothesis, but the present state of it, and the remarkable fact that mathematicians are in the midst of developing new methods by which the continuum hypothesis could be solved after all.

I will explain some of these developments, along with some of the more recent history of the continuum hypothesis, from the point of view of Kurt Gödel’s role in them. Gödel , a Member of the Institute’s School of Mathematics on several occasions in the 1930s, and then continuously from 1940 until 1976, 1 was a relative newcomer to the problem. But it turns out that Gödel’s hand is visible in virtually every aspect of the problem, from the post-Cantorian period onward. Curiously enough, this is even more true now than it was at the time of Gödel’s death nearly thirty-five years ago.

What is the Continuum Hypothesis?

Mathematics is nowadays saturated with infinity . There are infinitely many positive whole numbers 0, 1, 2, 3 . . . . There are infinitely many lines, squares, circles in the plane, balls, cubes, polyhedra in the space, and so on. But there are also different degrees of infinity. Let us say that a set—a collection of mathematical objects such as numbers or lines—is countable if it has the same number of elements as the sequence of positive whole numbers 1, 2, 3 . . . . The set of positive whole numbers is thus countable, and so is the set of all rational numbers. In the early 1870s , Cantor made a momentous discovery: the set of real numbers (such as 5, 17, 5/12, √–2, π, e, . . . ) sometimes called the “continuum,” is uncountable . By uncountable, we mean that if we try to count the points on a line one by one, we will never succeed, even if we use all of the whole numbers. Now it is natural to ask the following question: are there any infinities between the two infinities of whole numbers and of real numbers?

This is the continuum hypothesis, which proposes that if you are given a line with an infinite set of points marked out on it, then just two things can happen: either the set is countable, or it has as many elements as the whole line. There is no third infinity between the two.

At first, Cantor thought he had a proof of the continuum hypothesis; then he thought he could prove it was false; and then he gave up. This was a blow to Cantor, who saw this as a defect in his work—if one cannot answer such a simple question as the continuum hypothesis, how can one possibly go forward?

Some History

The continuum hypothesis went on to become a very important problem, so much so that in 1900 Hilbert listed it as the first on his list of open problems, as previously mentioned. Hilbert eventually gave a proof of it in 1925—the proof was wrong, though it contained some important ideas.

Around the turn of the century, mathematicians were able to prove that the continuum hypothesis holds for a special class of sets called the Borel sets. 2 This is a concrete class of sets, containing, for the most part, the usual sets that mathematicians work with. Even with this early success in the special case of Borel sets though, and in spite of Hilbert’s attempted solution, mathematicians began to speculate that the continuum hypothesis was in general not solvable at all. Hilbert , for whom nothing less than “the glory of human existence” seemed to depend upon the ability to resolve all such questions, was an exception. “ Wir müssen wissen . Wir werden wissen ,” 3 he said in 1930 in Königsberg . In a great irony of history, at the very same meeting, but on the day before, the young Gödel announced his first incompleteness theorem. This theorem, together with Gödel’s second incompleteness theorem, is generally thought to have dealt a death blow to Hilbert’s idea that every mathematical question that permits an exact formulation can be solved. Hilbert was not in the room at the time.

Gödel , however, became a strong advocate of the solvability of the continuum hypothesis, taking the view that his incompleteness theorems , though they show that some provably undecidable statements do exist, have nothing to do with whether the continuum hypothesis is solvable or not. Like Hilbert , Gödel maintained that the continuum hypothesis will be solved.

What is Provable Unsolvability Anyway?

We arrive at an apparent conundrum. On the one hand, the continuum hypothesis is provably unsolvable, and on the other hand, both Gödel and Hilbert thought it was solvable. How to resolve this difficulty? What does it mean for something to be provably unsolvable anyway?

Some mathematical problems may be extremely difficult and therefore without a solution up to now, but one day someone may come up with a brilliant solution. Fermat’s last theorem, for example, went unsolved for three and a half centuries. But then Andrew Wiles was able to solve it in 1994. The continuum hypothesis is a problem of a very different kind; we actually can prove that it is impossible to solve it using current methods , which is not a completely unknown phenomenon in mathematics. For example, the age-old trisection problem asks: can we trisect a given angle by using just a ruler and compass? The Greeks of the classical period were very puzzled by how to make such a trisection, and no wonder, for in the nineteenth century it was proved that it is impossible—not just very difficult but impossible. You need a little more than a ruler and compass to trisect an arbitrary angle—for example, a compass and a ruler with two marks on it.

It is the same with the continuum hypothesis: we know that it is impossible to solve using the tools we have in set theory at the moment. And up until recently nobody knew what the analogue of a ruler with two marks on it would be in this case. Since the current tools of set theory are so incredibly powerful that they cover all of existing mathematics, it is almost a philosophical question: what would it be like to go beyond set-theoretic methods and suggest something new? Still, this is exactly what is needed to solve the continuum hypothesis.

Consistency

Gödel began to think about the continuum problem in the summer of 1930, though it wasn’t until 1937 that he proved the continuum hypothesis is at least consistent . This means that with current mathematical methods, we cannot prove that the continuum hypothesis is false .

Describing Gödel ’s solution would draw us into unneeded technicalities, but we can say a little bit about it. Gödel built a model of mathematics in which the continuum hypothesis is true. What is a model? This is something mathematicians build with the purpose of showing that something is possible, even if we admit that the model is just what it is, a kind of artificial construction. Children build model airplanes; architects draw up architectural plans; mathematicians build models of the mathematical universe. There is an important difference though, between mathematicians’ models and architectural plans or model airplanes: building a model that has the exact property the mathematician has in mind, is, in all but trivial cases, extremely difficult. It is like a very great feat of engineering.

The idea behind Gödel’s model, which we now call the universe of constructible sets , was that it should be made as small as is conceivably possible by throwing everything out that was not absolutely essential. It was a tour de force to show that what was left was enough to satisfy the requirements of mathematics, and, in addition, the continuum hypothesis. This did not show that the continuum hypothesis is really true, only that it is consistent, because Gödel’s universe of constructible sets is not the real universe, only a kind of artifact. Still, it suffices to demonstrate the consistency of the continuum hypothesis.

Unsolvability

After Gödel’s achievement, mathematicians sought a model in which the continuum hypothesis fails, just as Gödel found a model in which the continuum hypothesis holds. This would mean that the continuum hypothesis is unsolvable using current methods. If, on the one hand, one can build a picture of the mathematical universe in which it is true, and, on the other hand, if one can also build another universe in which it is false, it would essentially tell you that no information about the continuum hypothesis is lurking in the standard machinery of mathematics.

So how to build a model for the failure of the continuum hypothesis? Since Gödel’s universe was the only nontrivial universe that had been introduced, and, moreover, it was the smallest possible, mathematicians quickly realized that they had to find a way to extend Gödel’s model, by carefully adding real numbers to it. This is hair-raisingly difficult. It is like adding a new card to a huge house of cards, or, more exactly, like adding a new point to a line that already is—in a sense—a continuum. Where do you find the space to slip in a few new real numbers?

Looking back at Paul Cohen ’s solution, a logician has to slap her forehead, not once, but a few times. His idea was that the real numbers one adds should have “no properties,” as strange as this may sound; they should be “generic,” as he called them. In particular, a Cohen real , as they came to be called, should avoid “saying anything” nontrivial about the model. How to make this idea mathematically precise? That was Paul Cohen’s great invention: the forcing method, which is a way to add new reals to a model of the mathematical universe.

Even with this idea, serious obstacles now stood in the way of a full proof. For example, one has to prove an extremely delicate metamathematical theorem—as these are called—that even though forcing extends the universe to a bigger one, one can still talk about it in the first universe; in technical terms, one has to prove that forcing is definable . Moreover, to violate the continuum hypothesis, we have to add a lot of new points to the continuum, and what we believe is “a lot” may in the final stretch turn out to be not so many after all. This last problem—the technical term is preserving cardinals —was a very serious matter. Cohen later wrote of his sense of unease at that point, “given the rumors that had circulated that Gödel was unable to handle the CH.” 4 Perhaps Cohen sensed, while on the brink of his great discovery, the almost physical presence of the one mathematician who had walked the very long way up to that very door, but was unable to open it.

Two weeks later, while vacationing with his family in the Midwest, Cohen suddenly remembered a lemma from topology (due to N. A. Shanin ), and this was just what was needed to show that everything falls into place. The proof was now finished. It would have been an astounding achievement for any set theorist, but the fact that it was solved by someone from a completely different field—Paul Cohen was an analyst after all, not a set theorist—seemed beyond belief.

Writing the Paper

The story of what happened in the immediate aftermath of Cohen’s announcement of his proof is very interesting, also from the point of view of human interest, so we will permit ourselves a slight digression in order to touch upon it here.

The announcement seems to have been made at a time when the extent of what had been shown was not clear, and the proof, though it was finished in all the essentials, was not in all details completely finished. In a first letter to Gödel , dated April 24, 1963, Cohen communicated his results. But about a week later, he wrote a second, more urgent letter, in which he expressed his fear that there might be a hidden flaw in the proof, and, at the same time, his exasperation with logicians, who could not believe that he was able to prove that very delicate theorem on the definability of forcing.

Cohen confessed in the letter that the situation was wearing, also considering “the unexpected interest my work has aroused among the general (non-logical) mathematical world.”

Gödel replied with a very friendly letter, inviting Cohen to visit him, either at his home on Linden Lane or in his office at the Institute, writing, “You have just achieved the most important progress in set theory since its axiomatization . So you have every reason to be in high spirits.”

Soon after receiving the letter, Cohen visited Gödel at home, whereupon Gödel checked the proof, and pronounced it correct.

What followed over the next six months is a voluminous correspondence between the two, centered around the writing of the paper for the Proceedings of the National Academy of Sciences . The paper had to be carefully written; but Cohen was clearly impatient to go on to other work. It therefore fell to Gödel to fine tune the argument, as well as simplify it, all the while keeping Cohen in good spirits. The Gödel that emerges in these letters—sovereign, generous, and full of avuncular goodwill, will be unfamiliar to readers of the biographies—especially if one keeps in mind that by 1963 Gödel had devoted a good part of twenty-five years to solving the continuum problem himself, without success. “Your proof is the very best possible,” Gödel wrote at one point. “Reading it is like reading a really good play.”

Gödel and Cohen bequeathed to set theorists the only two model construction methods they have. Gödel’s method shows how to “shrink” the set-theoretic universe to obtain a concrete and comprehensible structure. Cohen’s method allows us to expand the set-theoretic universe in accordance with the intuition that the set of real numbers is very large. Building on this solid foundation, future generations of set theorists have been able to make spectacular advances.

There was one last episode concerning Gödel and the continuum hypothesis. In 1972, Gödel circulated a paper called “Some considerations leading to the probable conclusion that the true power of the continuum is ℵ 2 ,” which derived the failure of the continuum hypothesis from some new assumptions, the so-called scale axioms of Hausdorff . The proof was incorrect, and Gödel withdrew it, blaming his illness. In 2000, Jörg Brendle , Paul Larson, and Stevo Todorcevic 5 isolated three principles implicit in Gödel’s paper, which, taken together, put a bound on the size of the continuum. And subsequently Gödel’s ℵ 2 became a candidate of choice for many set theorists, as various important new principles from conceptually quite different areas were shown to imply that the size of the continuum is ℵ 2 .

Currently, there are two main programs in set theory. The inner model program seeks to construct models that resemble Gödel’s universe of constructible sets, but such that certain strong principles, called large cardinal axioms, would hold in them. These are very powerful new principles, which go beyond current mathematical methods (axioms). As Gödel predicted with great prescience in the 1940s , such cardinals have now become indispensable in contemporary set theory. One way to certify their existence is to build a model of the universe for them—not just any model, but one that resembles Gödel’s constructible universe, which has by now become what is called “canonical.” In fact, this may be the single most important question in set theory at the moment—whether the universe is “like” Gödel’s universe, or whether it is very far from it. If this question is answered, in particular if the inner model program succeeds, the continuum hypothesis will be solved.

The other program has to do with fixing larger and larger parts of the mathematical universe, beyond the world of the previously mentioned Borel sets. Here also, if the program succeeds, the continuum hypothesis will be solved.

We end with the work of another seminal figure, Saharon Shelah . Shelah has solved a generalized form of the continuum hypothesis, in the following sense: perhaps Hilbert was asking the wrong question! The right question, according to Shelah , is perhaps not how many points are on a line, but rather how many “small” subsets of a given set you need to cover every small subset by only a few of them. In a series of spectacular results using this idea in his so-called pcf-theory , Shelah was able to reverse a trend of fifty years of independence results in cardinal arithmetic, by obtaining provable bounds on the exponential function. The most dramatic of these is 2 ℵω ≤ 2 ℵ0 + ℵ ω4 . Strictly speaking, this does not bear on the continuum hypothesis directly, since Shelah changed the question and also because the result is about bigger sets. But it is a remarkable result in the general direction of the continuum hypothesis.

In his paper, 6 Shelah quotes Andrew Gleason, who made a major contribution to the solution of Hilbert’s fifth problem:

Of course, many mathematicians are not aware that the problem as stated by Hilbert is not the problem that has been ultimately called the Fifth Problem. It was shown very, very early that what he was asking people to consider was actually false. He asked to show that the action of a locally-euclidean group on a manifold was always analytic, and that’s false . . . you had to change things considerably before you could make the statement he was concerned with true. That’s sort of interesting, I think. It’s also part of the way a mathematical theory develops. People have ideas about what ought to be so and they propose this as a good question to work on, and then it turns out that part of it isn’t so .

So maybe the continuum problem has been solved after all, and we just haven’t realized it yet.

1 Appointed to the permanent Faculty in 1953; 2 This was extended to the so-called analytic sets by Mikhail Suslin in 1917. Borel sets are named for Emile Borel, uncle of the late mathematician (and IAS Faculty member) Armand Borel.; 3 “We must know. We will know.”; 4 P. J. Cohen, “The Discovery of Forcing”; 5 In their "Rectangular Axioms, Perfect Set Properties and Decomposition"; 6 "The Generalized Continuum Hypothesis Revisited"

Some Mathematical Details

Intuitively, the set-theoretic universe is the result of iterating basic constructions such as products ∏ i ∈ I A i , unions U i ∈ I A i , and power sets P(A) . In addition, the universe is assumed to satisfy so-called reflection : any property that it has is already possessed by some smaller universe, the domain of which is a set. The process starts from some given urelements , objects that are not sets, i.e., do not consist of elements, but it has been proven that the urelements are unnecessary and the process can be started from the empty set. Iterating this process into the transfinite , we obtain the cumulative hierarchy V of sets. Transfinite iterations are governed by ordinals , canonical representatives of well-ordered total orders, denoted by lower-case Greek letters α , β , etc. The hierarchy V is defined recursively by V α = U β < α P ( V β ). The fact that V = U α V α is the entire universe of sets is the intuitive content of the axioms of Zermelo-Frankel set theory with the Axiom of Choice, or ZFC , the basic system we have been working with all along.

Now Gödel’s model of the ZFC axioms, the constructible hierarchy L = U α L α , where L α = U β <α P L ( V β ), is built up not by means of the unrestricted power set operation P ( A ), but by the restricted operation P L ( A ), which takes from P ( A ) only those sets that are definable in ( A , ∈). Gödel showed that we can consistently assume V = L , but Cohen showed that it is consistent to assume that there are real numbers that are not in L .

The Borel sets of reals are obtained from open sets by means of iterating complements and countable unions. If we enlarge the set of Borel sets by including images of continuous functions, we obtain the analytic sets; a set is coanalytic if its complement is analytic.

Finally, the projective sets are obtained from analytic sets by iterating complements and continuous images. The field of descriptive set theory asks, among other questions, whether the classical theory of analytic and coanalytic sets can be extended to the projective sets; in particular, whether the projective sets are Lebesgue measurable, and have the perfect set property and the property of Baire . This was settled in the 1980s with the work of Shelah and Woodin , building on earlier work of Solovay , who showed that the projective sets have these three properties as a consequence of the existence of certain so-called large cardinals. This also follows from projective determinacy , a principle that was shown by Martin and Steel to follow from the existence of such large cardinals. A cardinal α is called a large cardinal if V α behaves in certain ways like V itself. For example, in that case, V α is a model of ZFC , but more is assumed. A famous large cardinal is a measurable cardinal, introduced by Stanislaw Ulam , an example of which is the smallest cardinal that admits a nontrivial countably additive two-valued measure.

What a State Mathematics Would Be In Today . . .

Before coming to the Institute where he was appointed as one of its first Professors in 1933, John von Neumann was a student of David Hilbert’s in Göttingen . Von Neu­mann worked on Hilbert’s program to find a complete and consistent set of axioms for all of mathematics. In addition to his many other contributions to mathematics and physics, von Neumann defined Hilbert space (unbounded operators on an infinite dimensional space), which he used to formulate a mathematical structure of quantum mechanics. Below, the late Herman Goldstine , a former Member in the Schools of Mathematics, Natural Sciences, and Historical Studies, recalls von Neumann’s working dreams about Kurt Gödel’s incompleteness theorem(s). ( Ex­cerp­ted from an oral history transcript available at www.prince­ton.edu/%7Emudd/finding_aids/math­oral/pmc15.htm ; more information about von Neumann and Gödel is available at www.ias.edu/people/noted-figures .)

His work habits were very methodical. He would get up in the morning, and go to the Nassau Club to have breakfast. And then from the Nassau Club he’d come to the Institute around nine, nine-thirty, work until lunch, have lunch, and then work until, say, five, and then go on home. Many evenings he would entertain. Usually a few of us, maybe my wife and me. We would just sit around, and he might not even sit in the same room. He had a little study that opened off of the living room, and he would just sit in there sometimes. He would listen, and if something interested him, he would interrupt. Otherwise he would work away.

At night he would go to bed at a reasonable hour, and he would waken, I think, almost every night, judging from the things he told me and the few times that he and I shared hotel rooms. He would waken in the night, two, three in the morning, and would have thought through what he had been working on. He would then write. He would write down the things he had worked on. . . .

He, under Hilbert’s tutelage, was trying to prove the opposite of the Gödel theorem. He worked and worked and worked at this, and one night he dreamed the proof. He got up and wrote it down, and he got very close to the end. He went and worked all day on that part, and the next night he dreamed again. He dreamed how to close the gap, and he got up and wrote, and he got within epsilon of the end, but he couldn’t make the final step. So he went to bed. The next day he worked and worked and worked at it, and he said to me, “You know, it was very lucky, Herman, that I didn’t dream the third night, or think what a state mathematics would be in today.” [Laughter.]

Juliette Kennedy is Associate Professor in the Department of Mathematics and Statistics at the University of Helsinki and a Member (2011–12) in the School of Historical Studies. In the history and foundations of mathematics, she has worked extensively on a project that attempts to put Kurt  Gödel  in full perspective, historically and foundationally. Her project at the Institute this year is centered on  Gödel’s  notion of semantic content. The mathematical aspect of the project involves the question of how many of the larger “large cardinals” can be captured with a newly discovered class of L-like inner models of set theory.

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Can the continuum hypothesis be settled in physics?

Can the continuum hypothesis be settled in physics? In a lecture mathematician Woodin considers the possibility :

Develops the mathematical physics of a mathematical understanding of the physical universe. If it starts to need large Cardinals remember large carnal axioms with finite-istic consequences so it's not completely unreasonable that large Cardinal axioms provide mathematical truths that you need to do the analysis of physics we already saw an instance of that infinitely many wooden Cardinals imply that the projective continuum hypothesis is true that's a remarkable connection between very large sets and very small sets who's to say that doesn't happen in physics somewhere so that would be a win and I'll tell you I would be as stunned.

What are philosophers take on such kind of claims? (feel free to include references) Are they genuinely considered?

• metaphysics
• philosophy-of-mathematics

• Comments are not for extended discussion; this conversation has been moved to chat . –  Geoffrey Thomas ♦ Commented Jan 30, 2023 at 9:15
• Cough cough ... Can an answer be updated? ;p –  More Anonymous Commented Mar 20 at 10:38
• An answer can be edited and updated. This is unrelated to comments being moved to Chat. –  Geoffrey Thomas ♦ Commented Mar 21 at 8:59

4 Answers 4

Physics is not about this sort of thing..

In physics you make observations. You then try to write down equations that describe these observations. The simpler the equations the better. These equations are called a model.

Physics can only settle an issue by providing a model that does not break when we plug in extra data. The closest physics can come to settling the continuum hypothesis is to create a mathematical model that requires we assume the existence of a set of a given infinite size, and that model is somehow better than those without the assumption.

For an easier example consider how most models contain the set of real numbers ℝ in some form. Space is assumed to be continuous. It is a product of real lines. This allows us to integrate the forces on a particle over a given path and get the change in energy. The change in energy and the forces are all made of real numbers.

These theories work. Does that settle the issue of whether the real numbers exist? Or does it merely imply they are a good mathematical convenience?

Perhaps there is a better model where ℝ is replaced with some exotic type of ordered set with more elements than the natural numbers but fewer than ℝ itself. Would such a model prove the continuum hypothesis?

Even the existence of infinity is tenuous in physics. The infinite sets that occur there are all sets of infinitely many possibilities . There are continuum-many values the energy of a particle could take. But we have only ever measured finitely many of those values.

From a mathematical point of view, it would be unlikely to find a box containing uncountably many particles. Countably many perhaps. But if we add up the contribution of uncountably many things it ends up that all but countably many are zero.

The main philosophical positions here are the various versions of mathematical platonism and...those who aren't platonist, like logicism and formalism . But really the relevant distinction here is the distinction between realism and non-realism (or anti -realism):

Do we believe that mathematical abstract objects (such as sets) have an existence independent of the human mind?

Platonists - and realists in general - say "yes", but they differ greatly in what exactly such "existence" means. For example in re structuralists would probably find the suggestion that mathematical statements about sets can be investigated by looking at the physical world rather plausible, since they have to believe that there are physical structures exemplifying all mathematical constructs.

Formalists, logicists and other non-realists would find the suggestion absurd: To them, mathematical objects and statements have no existence independent of the formulae used to describe them; the "truth" of a mathematical statement is not a statement about anything that exists at all, but statements about what can be abstractly derived given a particular axiom set and a particular set of rules of deduction.

Concerning set theory generally and statements about infinite sets like the continuum hypothesis specifically, realists run into a problem similar to the problem of induction in the philosophy of science: How does one, in practice, exhibit an infinite set or a map between infinite sets? The structuralist who believes there must be physical structures exemplifying an infinite set can, by the nature of infinity, never even finish recounting the members of even one of these sets!

Not all realists despair at this problem, but a notable fraction become (ultra)finitists who do believe in the existence of mathematical objects but not in infinite objects.

• I think the question is about mathematical structure of theories in physics and not about concrete objects in the nature. As an example, axion of choice is equivalent to the existence of a basis for infinite dimensional Hilbert spaces, so one can argue that infinite dimensional Hilbert spaces are necessary for quantum physics and as a result, physics can provide some evidence for truth of axiom of choice. There may be some similar story for continuum hypothesis. –  Arian Commented Jan 29, 2023 at 18:38
• 1 @Arian 1. The axiom of choice and the existence of infinite-dimensional orthonormal bases are not known to be equivalent - AC implies the existence of the basis, but the converse is not known, cf. this math.SE question . 2. You are implicitly taking a realist view here if you think that the phrase "evidence for truth of axiom of choice" actually means something. To a non-realist, this just doesn't make any sense - the axiom of choice is not a claim about reality, so it cannot have "evidence" for or against it. –  ACuriousMind Commented Jan 29, 2023 at 19:39
• 1 (The idea that because the physical theory that models reality requires the axiom of choice this constitutes "evidence" for the axiom of choice is close to a in re structuralist view that reality then "exemplifies" the axiom of choice) –  ACuriousMind Commented Jan 29, 2023 at 19:40
• Thanks, I mean Blass theorem and I mistakenly referred to it. You should take truth as correspondence and then say I am a realist. I also remembered that there are some papers on continuum hypothesis and some model of hidden variable quantum mechanics, arxiv.org/abs/1212.0110 . –  Arian Commented Jan 29, 2023 at 20:12
• @Arian how important are bases for quantum mechanics? –  Daron Commented Jan 29, 2023 at 21:38

The continuum hypothesis is already settled. The answer to it is that there are models of set theory in which the continuum hypothesis is true, and there are also models of set theory in which the continuum hypothesis is false. Both kinds of models are fully valid and self-consistent; there is nothing wrong with either kind of model.

By analogy, is up positive and down negative, or is down positive and up negative? Both systems are fully valid and self-consistent; there is nothing wrong with either one.

That said, there's still a meaningful question here: does physics provide us with some reason to favor one kind of model of set theory over the other? Is one kind better or more useful than the other one when we're talking about physics?

So far, the answer seems to be no. As far as I know, there is no obvious connection between the mathematics describing the laws of physics, and the question of how many cardinalities there are among infinite sets of real numbers. If one mathematician says "yes" to the continuum hypothesis and another one says "no," then neither mathematician will have an easier or harder time with physics as a result of that choice.

Perhaps in the future, some new laws of physics will be discovered that are related to cardinalities of infinite sets of real numbers, and we will find that these laws of physics are particularly easy to describe if we take the continuum hypothesis to be true, but hard to describe if we take the continuum hypothesis to be false—or the opposite. (My gut feeling says that that's pretty unlikely.)

Even if the laws of physics turn out to favor models of set theory where the continuum hypothesis is true (or false), that won't mean that in our universe, the continuum hypothesis is true (or false). One of the two options may turn out to be less useful for physics, but being less useful doesn't mean that it's wrong.

• Indeed CH holds in some models and not others. I interpret the question to be about whether or not CH holds in the universe of all physical objects. –  Daron Commented Jan 31, 2023 at 1:40

One underdeterminate epistemic possibility in this connection is that a theory about perception itself might involve making a judgment about the cardinality of the set of real numbers. The open coloring question , for example, might conceivably be relevant to a theory of color perception (albeit quite in abstracto ), and since two samples of open-coloring axioms in tandem resolve the natural powerset to ℵ 2 , one might reason from an attendant theory of color perception (if it's actually possible/relevant) to the conclusion that the natural powerset is ℵ 2 .

Another (not-so-clear) possibility might be that forcing as a mathematical phenomenon has a physical counterpart, and this counterpart can change the size of physical continua . I've been working on an attempt to model laws-of-physics on infinite conjunctions in infinitary logic where for t (time) = n , our physical universe has an infinitary logical signature ℒ(ω f ( n ) , ω g ( n ) ) such that some functions f and g yield an evolution of the physical world's logical signature over time (this is to try to implement Lee Smolin's changing-laws-of-physics idea ), but to be honest, I haven't gotten much further than a nifty background for a science fiction storyline, not a genuine scientific hypothesis. (I.e., what predictive value does this "model" have, if any? The best I've thought of would be that our capacity for continuous perception would change with the changing cardinality of physical continua, but how would that be "noticeable" or evaluable, then?)

Broadly, one problem with thinking that mathematical physics, much less experimental physics that's mathematically informed, would be amenable to novel reasoning for or against CH is that there are an extremely vast number of alternatives to CH in higher set theory. ZFCwise, there are absolutely infinitely many alephs that the natural powerset can be forced to equal, and beyond the edge of ZFC, there is even the possibility of forcing the Continuum to equal absolute infinity itself (see also Timothy James' essay on predicativism in the philosophy of mathematics on the "indefinite extensibility" of the natural powerset; or consider that the surreal number line itself contains absolutely infinitely many infinitesimals in every interval, incl. [0, 1]). Perhaps physics would at least allow us to eliminate the prior disjunct in {CH ∨ ~CH} but the latter disjunct is so internally vast that said elimination would be as meager a contribution to the issue as possible. Even worse, it's not only that the natural powerset can be forced to equal so many things on its own, but: the powerset of the zeroth aleph can be forced to equal the powerset of the first aleph, as well as the second, third, fourth, etc. alephs , and so indeed, modulo the proper-class scales of options, we might force every well-ordered transfinite cardinal, prior to ℶ 1 , to equal ℶ 1 , so that the first beth is a fixed point of the aleph function . ℝ On the surface, it is hard to say how empirical information, or mathematical models of said information, would include strong, clear reasons for filtering in, or out, so many options.

ℝ Even more insidiously, suppose that the well-ordering principle is waived (because the basic axiom of choice, or whichever choice axiom, is waived). Then it is possible for the cardinality of the Continuum to be a transfinite cardinal, but not from the well-ordered sequence of such cardinals, i.e. it would not be an aleph but perhaps similar to (apparently not identical to, though) an amorphous set .

Perhaps most insidiously of all, suppose you waive the powerset axiom itself. You can still use the classical diagonal argument to show that the set of real numbers is not bijective with the set of natural numbers, but you no longer have that 2 ℵ 0 is the arithmetical expression of this difference. (I don't think this is plausible at all, since I think John Conway's explanation for the exponential expression fitting the case is a perfectly apparent explanation, but on the other hand, see again James' predicativist apologetics for how the exponential function on whichever X can come apart from the concept of "the set of all subsets of X .")

EDIT: Though this answer was accepted and has received a number of upvotes, it was also downvoted, and I feel like I worded it in an imprecise way. Firstly, on the "yes" side, my references to open coloring axioms and physical forcing are very speculative; insofar as reductionism is not in vogue anymore, I imagine that biological/neurological theories of color perception might indeed be relevant to physical theories in a way that could also play into evaluating some version of the Continuum problem, but I am not especially well-versed in actual physics, biology, or neurology, so I feel like I should emphasize just how speculative my comments on this score are.

Second, on the "maybe not" side: overall, I do not think that there is really just one powerset function. Cantor's theorem has it that various sets of subsets must exceed their bases, but complications involving definable vs. hyperdefinable (or even antidefinable) subsets seem to multiply the question of such a function. I think that the enduring desire to "settle" the Continuum issue is often derived from the seeming continuity involved in physical perception, so that, "How many points are there in physical space?" seems like a question of external/objective reality. So rather than say that every powerset function might be resolved by some future theory of physics, I would rather say that something like "the set of all physically realizable subsets of a countable set" would be the specific subtype of the powerset function whose size could be "settled" in such a theory. But this too is speculative, after all; I offer these conjectures as an answer to the OP question in only the bare sense that they address the modal term in that question per the title of the OP post: i.e., in some abstract sense of "can," CH, or a version of it anyway, "can" be settled by physics.

Finally, there is one more "insidious" variation on the "maybe not" theme that comes to my mind, one based on paraconsistent set theory . In PST, perhaps, one might force the Continuum's cardinality to equal several inconsistent numbers at once; applying this "model" to physics, or physics to this "model," would then mean bringing in considerations like a paraconsistent theory of superposition . Paraconsistent logic is motivated by the desire to avoid an inferential explosion, and forcing the Continuum's cardinality to equal every option otherwise delineated above is almost the same as (or maybe even identical to, eventually) such an explosion; so a paraconsistent set theorist would still be motivated to include some restrictions on their "insidious" standpoint, to rule out the most deviant alternative to CH of all.

But so again, overall, for a theory of physics to settle any version of the Continuum question would mean that such a theory would have to sort through questions about choice axioms, forcing, the geometry of perception, etc., all in a way that comports with how theories of physics are reliably set up. I'm not a physicist and I am reluctant to press my claims, here, too strongly, out of concern for veering off into pseudoscientific territory. I appreciate that my answer was accepted, and I think my answer involves informed reflection on the parameters of the OP question, but insofar as all this is philosophical reflection, I am still highly uncertain about my conclusions in this case.

• @user4894 off the top of my head, I don't know of any "obviously philosophical" essays about positively using physics theories to address CH. However, against using such theories in that way, well... that's where the greater weight of my post lies. Predicativism, for example, is a school of philosophy of math that brings into sharp relief some of the issues with portraying CH as an objective matter, and if CH is not so objective, it will perhaps be harder to link its settlement to more objectively-minded theories of physics. –  Kristian Berry Commented Jan 29, 2023 at 22:50
• as you are, per your edit, concerned with the downvotes, here's my reason: while I cannot judge the quality of your question, I can confidently say that unless a reader is already deeply familiar with the very specific topics you bring up, he won't understand much of it, nor be able to bring it in contact with the question. The other two answers show how a less deep level of detail can answer the question just as well, with the added benefit of average readers (who are not deep into mathematics of infinity) being able to understand them relatively easily. –  AnoE Commented Jan 30, 2023 at 13:17
• @AnoE fair enough, except maybe not-upvoting would be more fair than actively-downvoting. I know this is the PhilosophySE, not the PhilosophyOverflow (and there isn't a PhilosophyOverflow), but so I still provided links for pretty much every technical detail I brought up, and I don't see why answers with fewer citations would really be better fits for the SE network as whole than one which pools together a great deal of relevant information. –  Kristian Berry Commented Jan 30, 2023 at 20:40
• @KristianBerry i am considering removing your answer from the accepted answer because of it might be bringing unwanted attention. I also suspect many people downvote when they do not agree (but in that case i would recommend commenting or answering while mentioning). If ur okay with the downvotes onslaught I'll let it remain? –  More Anonymous Commented Jan 31, 2023 at 4:28
• @MoreAnonymous yeah you can unaccept my answer, if I can think of improvements to my answer I will make them but for the time being I don't think my answer is helpful. Perhaps I will just delete the answer altogether. –  Kristian Berry Commented Jan 31, 2023 at 8:40

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Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis ( There does not exist a set with a cardinality less than the reals and no set strictly greater than the natural numbers. ) is neither true or false.

This is utterly baffling to me, If it's possible to construct a set between $\mathbb{N}$ and $\mathbb{R}$ then this statement is demonstrably false, but if not then the statement is true.

This seems to be a straitforward deduction, but many with a more advanced understanding of the topic matter believe CH to be neither.

How can this be?

• incompleteness

• 10 $\begingroup$ A related previous question: "Impossible to prove" vs "neither true nor false" $\endgroup$ –  user856 Commented Sep 1, 2012 at 0:12
• 2 $\begingroup$ This type of thing makes me like intuitionistic logic more. $\endgroup$ –  Tunococ Commented Sep 1, 2012 at 0:14
• 4 $\begingroup$ "If it's possible to construct a set between $\mathbb{N}$ and $\mathbb{R}$ then this statement is demonstrably false, but if not then the statement is true." The predominant attitude among modern mathematicians is that math doesn't have to be constructive: that we can prove things exist without constructing them explicitly, and that such a proof is satisfactory. It then becomes possible to have things that you can prove exist, and can also prove can't be constructed explicitly; this is the situation, e.g., with ultrafilters. Independence of CH from ZFC says we can't rule this out for CH. $\endgroup$ –  user13618 Commented Sep 2, 2012 at 14:54

7 Answers 7

Set theory is much more complicated than "common" mathematics in this aspect, it deals with things which you can often prove that are unprovable.

Namely, when we start with mathematics (and sometimes for the rest of our lives) we see theorems, and we prove things about continuous functions or linear transformations, etc.

These things are often simple and have a very finite nature (in some sense), so we can prove and disprove almost all the statements we encounter. Furthermore it is a good idea, often, to start with statements that students can handle. Unprovable statements are philosophically hard to swallow, and as such they should usually be presented (in full) only after a good background has been given.

Now to the continuum hypothesis. The axioms of set theory merely tell us how sets should behave. They should have certain properties, and follow basic rules which are expected to hold for sets. E.g., two sets which have the same elements are equal.

Using the language of set theory we can phrase the following claim:

If $A$ is an uncountable subset of the real numbers, then $A$ is equipotent with $\mathbb R$.

The problem begins with the fact that there are many subsets of the real numbers. In fact we leave the so-called "very finite" nature of basic mathematics and we enter a realm of infinities, strangeness and many other weird things.

The intuition is partly true. For the sets of real numbers which we can define by a reasonably simple way we can also prove that the continuum hypothesis is true: every "simply" describable uncountable set is of the size of the continuum .

However most subsets of the real numbers are so complicated that we can't describe them in a simple way. Not even if we extend the meaning of simple by a bit, and if we extend it even more, then not only we will lose the above result about the continuum hypothesis being true for simple sets; we will still not be able to cover even anything close to "a large portion" of the subsets.

Lastly, it is not that many people "believe it is not a simple deduction". It was proved - mathematically - that we cannot prove the continuum hypothesis unless ZFC is inconsistent, in which case we will rather stop working with it.

Don't let this deter you from using ZFC, though. Unprovable questions are all over mathematics, even if you don't see them as such in a direct way:

There is exactly one number $x$ such that $x^3=1$.

This is an independent claim. In the real numbers, or the rationals even, it is true. However in the complex numbers this is not true anymore. Is this baffling? Not really, because the real and complex numbers have very canonical models. We know pretty much everything there is to know about these models (as fields, anyway), and it doesn't surprise us that the claim is true in one place, but false in another.

Set theory (read: ZFC), however, has no such property. It is a very strong theory which allows us to create a vast portion of mathematics inside of it, and as such it is bound to leave many questions open which may have true or false answers in different models of set theory. Some of these questions affect directly the "non set theory mathematics", while others do not.

Some reading material:

• A question regarding the Continuum Hypothesis (Revised)
• Neither provable nor disprovable theorem
• Impossible to prove vs neither true nor false

• 3 $\begingroup$ It's always fun when everyone writes a short answer and I find myself writing a short story! :-) $\endgroup$ –  Asaf Karagila ♦ Commented Sep 1, 2012 at 0:59
• 7 $\begingroup$ This answer is illuminates a murky subject matter with exceptional clarity and wisdom. Thank you! $\endgroup$ –  zetavolt Commented Sep 1, 2012 at 1:16
• 12 $\begingroup$ "...the real and complex numbers have a very canonical model. We know pretty much everything there is to know about these models..." except for whether they have any subsets with cardinality strictly between that of the integers and the reals. And where to find the zeros of the zeta function. $\endgroup$ –  Gerry Myerson Commented Sep 1, 2012 at 1:21
• 2 $\begingroup$ @Mark: I think that writing a +17 answer about independence of CH from ZFC should hint that I am probably aware to that, among other things such as the :-) , however for the general point for those who might be unaware of this fact: thanks. $\endgroup$ –  Asaf Karagila ♦ Commented Sep 1, 2012 at 7:57
• 2 $\begingroup$ @Andrew: "Continuum" is a term for the real numbers. It has other, more general meanings, but in this context it simply means the real numbers. $\endgroup$ –  Asaf Karagila ♦ Commented Apr 21, 2014 at 19:50

A number of mathematicians have definite opinions about the truth of CH, the majority I believe opting for false, Kurt Gödel among them. What there is agreement on, because it is a theorem, is that CH is neither provable nor refutable in ZFC. But that is quite a different assertion than "neither true nor false."

The theory ZFC captures many common intuitions about sets. It has been the dominant "set theory" for many years. There is no good reason that it will remain that forever.

• 1 $\begingroup$ Doesn't one have to say, "neither provable nor refutable in ZFC, provided ZFC is consistent"? $\endgroup$ –  Gerry Myerson Commented Sep 1, 2012 at 1:16
• 2 $\begingroup$ @GerryMyerson: Yes. But the axioms (at least of ZF) are true in the universe of sets, and the relative consistency of ZFC is a theorem. So I thought that in the context of a question about truth, the obligatory "if ZFC is consistent" could be dispensed with. $\endgroup$ –  André Nicolas Commented Sep 1, 2012 at 1:20
• $\begingroup$ Would you know (trusted estimates of) the proportion of mathematicians with definite opinions about the truth of CH , and amongst them, the proportion of those opting for false ? $\endgroup$ –  Did Commented Sep 1, 2012 at 14:23
• 2 $\begingroup$ Should probably not have said mathematicians. For set theorists, one can make lists. The pro-CH list would be short. $\endgroup$ –  André Nicolas Commented Sep 1, 2012 at 17:58
• $\begingroup$ WP has a nice summary of this: en.wikipedia.org/wiki/… $\endgroup$ –  user13618 Commented Sep 2, 2012 at 14:46

One can construct a model of set theory in which CH is true, and one can construct a model in which CH is false.

• $\begingroup$ I think with and without axiom of choice? how is it? can you give a bit more info? thanks. $\endgroup$ –  Seyhmus Güngören Commented Sep 1, 2012 at 0:11
• $\begingroup$ Can you elaborate on this answer? $\endgroup$ –  zetavolt Commented Sep 1, 2012 at 0:14
• $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ –  Robert Israel Commented Sep 1, 2012 at 0:16
• $\begingroup$ @SeyhmusGüngören: The axiom of choice has nothing to do with this... $\endgroup$ –  Asaf Karagila ♦ Commented Sep 1, 2012 at 0:21
• 2 $\begingroup$ Asaf, I think that's what @Seyhmus was asking: there are models both with and without Choice in which CH is true, and models both with and without Choice in which CH is false. $\endgroup$ –  Gerry Myerson Commented Sep 1, 2012 at 1:14

It is not possible to explicitly "construct" such a set and prove (using the ZFC axioms of set theory) that its cardinality is strictly between those of $\mathbb N$ and $\mathbb R$. That doesn't mean that no such set exists.

As a Platonist, I would not say that CH is "neither true nor false", rather that we do not know (and in a certain sense we cannot know) which it is. Truth and provability are very different things.

• 5 $\begingroup$ We can actually prove that it is true or false... $\endgroup$ –  Michael Greinecker Commented Sep 1, 2012 at 0:18
• $\begingroup$ @MichaelGreinecker, could you explain that? $\endgroup$ –  user56834 Commented Jan 23, 2018 at 19:31
• 3 $\begingroup$ @Programmer2134 For every statement $A$, $A\vee\neg A$ is a true statement in classical logic. But that does not mean there is a proof of $A$ or a proof of $\neg A$. $\endgroup$ –  Michael Greinecker Commented Jan 23, 2018 at 20:31
• $\begingroup$ That assumes you can have this as a statement $\endgroup$ –  Gabriel Tellez Commented Jun 2 at 1:10
• $\begingroup$ @GabrielTellez I'm not sure what your objection is. CH is a statement of set theory. $\endgroup$ –  Robert Israel Commented Jun 3 at 2:09

If you take the parallel axiom away from Euclidean geometry, you cannot prove (using the remaining axiom system) whether it is true or false. But even in a geometry without parallel axiom, you can have interesting results (see http://en.wikipedia.org/wiki/Absolute_geometry ).

• 1 $\begingroup$ Maybe add a sample of said interesting results . $\endgroup$ –  Did Commented Sep 1, 2012 at 11:39
• 1 $\begingroup$ So you added a link to a WP page on Absolute geometry. Fine. But could you be more specific about interesting results ? $\endgroup$ –  Did Commented Sep 2, 2012 at 12:53
• 1 $\begingroup$ WP mentions the first 28 propositions from Euclid's Elements, the exterior angle theorem and the Saccheri-Legendre theorem. Maybe this isn't terrible interesting, but I guess good enough to get my point across. It's just an analogy, dude... $\endgroup$ –  Landei Commented Sep 2, 2012 at 13:18

I beleive that your confusion rise from bad definition of CH. It is not "there does not exist a set with a cardinality less than the reals and no set strictly greater than the natural numbers" as you stated it, but rather "there does not exist a set with a cardinality less than the reals AND strictly greater than that of the natural numbers.".

(1)To make your wording more accurate, I assume you mean "there exists" when you state it as "It's possible to construct."

(2)Truth and Falsity in logic might have a different meaning than you think. Truth and Falsity here can only be talked within a scope, which is given by ZFC axiom system. Under the consistency assumption of ZFC, the system is incomplete, meaning that there are statements there not having a T/F answer.

(3) When people say CH "is neither true or false" in this context, they really just mean that such Truth or Falsity cannot be deduced from ZFC system. More precisely, they mean that under the consistency assumption of ZFC, if you add CH or its negation to ZFC, the system remains consistent, and therefore CH and its negation cannot be deduced from ZFC.

(4) In a larger system, it is possible to give a definitive answer to CH. As a trivial example, if we add CH to ZFC, then CH would be true.

• $\begingroup$ You missed "HC" in the last sentence of the 3rd point. $\endgroup$ –  Asaf Karagila ♦ Commented Nov 18, 2014 at 21:25
• $\begingroup$ Thank you. I did misspelled it. $\endgroup$ –  Deadwood Kumu Commented Nov 18, 2014 at 23:32

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