Abstract
The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different argument, an argument that Cantor’s hypothesis is in fact true. This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets.
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Notes
Nonplatonist conceptions of the metaphysics of set theory are dismissed by Woodin as basically untenable and are not discussed in this paper. See Woodin (2009a) for Woodin’s argument against nonplatonism.
Remark: it is unknown, in fact unprovable from the \(ZFC\) axioms, if \(ZFC\) is consistent. The same holds true for number theory, for example. However, experience with these axiom systems makes it reasonable to expect that these systems are in fact consistent. It is common practice in set theory to assume the consistency of \(ZFC\), and Woodin does so as well: “There will be no discovery ever of an inconsistency in \(ZF+AD\)” (Woodin 2009b, p. 10) (note that the consistency of \(ZF\) implies the consistency of \(ZFC\) by Gödel’s construction of \(L\)). I shall follow Woodin in this throughout this paper.
There is one exception to this. Kunen has proved that the existence of a Reinhardt cardinal is inconsistent with \(ZFC\). The proof relies on the Axiom of Choice, and it is unknown if the existence of a Reinhardt is inconsistent with \(ZF\) (i.e. \(ZFC\) minus the Axiom of Choice). See Jech (2006) for details.
Consistency strength: theory \(T\) has a higher consistency strength than theory \(S\) if the consistency of \(T\) implies the consistency of \(S\).
The cumulative hierarchy incorporates the idea that the universe of sets is buildup from below by taking the powerset and the union operation. This results in ‘levels’ within the hierarchy. See also the definition of the von Neumann universe \(V\) in Sect. 6.
Woodin remarked that it is theoretically possible that a generalisation of forcing could be found, which could be used to prove independence results about PA. But not even an idea of how to come by such a generalisation, let alone the generalisation itself, is currently present.
For a formal definition, see Sect. 6.
A set \(a\) is transitive if from \(b\in a\) and \(c\in b\) follows that \(c\in a\). Transitive sets turn out to be of particular interest to the set theorist. The transitive closure of a set \(x\) is the smallest transitive set, which contains \(x\).
Secondorder languages allow quantifications not only over variables, but also over sets of variables.
A proper class is a collection, which is too big to be a set. Classical examples of a proper classes are the universe of all sets and the collection of all the ordinal numbers. Classes are too big to be measured by standard measurements via cardinalities. A collection is said to contain class many elements if it contains as many elements as there are ordinals.
Note that Woodin’s \(\Omega \)logic is different from \(\omega \)logic.
\(A\)logic is sound if from the fact that there is an \(A\)proof of the statement \(\phi \) from some premisses \(T\), i.e. \(T\vdash _A\phi \), it follows that \(\phi \) is also \(A\)valid from \(T\), i.e. \(T\models _A\phi \). \(A\)logic is complete if \(T\models _A\phi \) implies that \(T\vdash _A\phi \). From Gödel’s Completeness Theorem, it follows that for classical logic, \(T\models \phi \) iff \(T\vdash \phi \).
For connoisseurs: To demand unboundedly, many Woodins is to demand a process. That this process can be regarded as an object in its own right is due to a principle already present in Cantor: ‘for every rule or process by means of which a collection of elements is obtained there is a set which contains exactly the elements which conform to the rule, or are obtained in the process, respectively’ (Fraenkel et al. 1984, pp. 30–31). Of course today we know that the above formulation leads to inconsistencies (it allows for the Russell set) and have hence changed the word ‘set’ to ‘class’ in the above. The process of having unboundedly many Woodin cardinals is hence simply the same as having a class of Woodin cardinals.
I am indebted to José Ferreirós and especially Dominik Adolf for drawing my attention to this subtlety.
For connoisseurs: Given a universally Baire set \(A\), an \(\epsilon \)model \(M\) of (a fragment of) \(ZFC\). Then, \(M\) is \(A\)closed if for all posets \({\mathbb {P}}\in M\) and all \(V\) generic filters \(G\subseteq {\mathbb {P}}\):
\(V[G]\models M[G]\cap A_G \in M[G]\).
For connoisseurs: There are only countably many sentences in the language of set theory, and the universally Baire sets are closed under preimages by Borel functions and countable unions. Therefore, there exists a single \(A_\Omega \) such that the class of models under consideration here are the \(A_\Omega \)closed models.
Keep in mind that the existence of a proper class of Woodin cardinals guarantees PD.
For connoisseurs: The assumption is needed for \(\Omega \)consistency.
See Sect. 6 for a discussion of \(L({\mathbb {R}})\).
This argument can be strengthened by regarding the corresponding claim for the nontrivial \(\Omega \)satisfiability of the \(\Omega \)conjecture, see (Woodin 2010b, p. 4).
In the ordinary construction of \(L\), we use the definable powerset operation \(\mathcal {P}_{Def}\) for successor steps: \(L_{\alpha +1}=\mathcal {P}_{Def}(L_\alpha )\), where \(L_{\alpha +1}\) contains all those subsets of \(L_\alpha \) that are definable from a finite number of elements of \(L_\alpha \) by a formula relativised to \(L_\alpha \). For the construction of \(L[E]\), not only the elements of \(L_\alpha \) are used as defining parameters, but also elements of \(E\). The construction of \(L(E)\) proceeds just like the construction of \(L\) but rather than starting form the empty set \(\emptyset \) the construction is started from the set \(E\). See Sect. 6 for formal definitions. The differences between \(L[E]\) and \(L(E)\) are most clearly seen by observing that, in general, \(E\in L(E)\) but not \(E\in L[E]\).
Notice one key difference in the assumptions between Gödel’s construction of a model and how the inner model programme resolves the issue. Gödel did not assume anything about \(AC\) or \(CH\), yet his resulting model believed both statements. Hence, he had proven consistency of \(AC\) and \(CH\) with \(ZF\). The inner model programme on the other hand assumes the consistency of \(ZFC\) plus whatever large cardinal axiom is under scrutiny and then tries to find a model for it. That is, the inner model programme does not prove consistency of \(ZFC\) plus the large cardinal axiom because consistency is already set as a premiss.
The reader might find it helpful to refer back to Sect. 2 and the (incomplete) list of large cardinal axioms given there to ‘see the picture’.
See Sect. 6 for a formal definition.
For connoisseurs: the concept of a strategic extender model is an evolution of Jensen’s and Dodd’s concept of a mouse. I am indebted to Dominik Adolf for pointing this out to me.
This term was coined in Koellner (2013).
I am indebted to Dominik Adolf for this proof.
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Acknowledgments
I would like to thank Brendan Larvor, Dominik Adolf, Jeremy Grey and José Ferreiós for their helpful comments.
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Communicated by : Jeremy Gray.
Appendices
Appendix A: Large cardinal axioms
This appendix gives definitions of all the large cardinal axioms mentioned in Sect. 2. All of the definitions are minor reformulations of the definitions given in Jech (2006) and Kanamori (2009). The latter is considered to be a classical text on the historical development of large cardinal axioms. A good and short introduction to the topic is Honzik (2013).
An elementary embedding, \(j\), is a truthpreserving function between two models. The critical point of \(j\), \(crit(j)\), is the smallest ordinal \(\alpha \) such that \(\alpha \ne j(\alpha )\). A simple proof by transfinite induction shows that \(crit(j)<j(crit(j))\), and it can be shown that \(crit(j)\) is always a cardinal. The identityembedding is the trivial elementary embedding and in all the definitions that follow \(j\) is nontrivial.
Definition 5.1
A Reinhardt cardinal is a \(\kappa =crit(j)\) for some elementary embedding \(j: V \rightarrow V\).
Definition 5.2
An nhuge cardinal is a cardinal \(\kappa \) such that there exists an elementary embedding \(j: V \rightarrow M\) with critical point \(\kappa \) such that \(M^{j^{n}(\kappa )}\subset M\).
Definition 5.3
A huge cardinal is a cardinal \(\kappa \) such that there is an elementary embedding \(j: V \rightarrow M\) with critical point \(\kappa \) such that \(M^{j(\kappa )}\subset M\).
Definition 5.4
An extendible cardinal \(\kappa \) is such that for every \(\alpha >\kappa \) there is an ordinal \(\beta \) and an elementary embedding \(j: V_\alpha \rightarrow V_\beta \) with critical point \(\kappa \).
Definition 5.5
A supercompact cardinal is an uncountable cardinal \(\kappa \) such that for every \(A\) with cardinality greater or equal to \(\kappa \) there exists a normal measure on \(P_\kappa (A)=\{X\subset A\mid \left X \right <\kappa \}\).
Definition 5.6
A superstrong cardinal is a cardinal \(\kappa \) such that there exists a elementary embedding \(j: V \rightarrow M\) with critical point \(\kappa \) such that \(V_{j(\kappa )}\subset M\).
Definition 5.7
A Woodin cardinal is a cardinal \(\delta \) such that for all \(A\subset V_\delta \) there are arbitrarily large \(\kappa <\delta \) such that for all \(\lambda <\delta \) there exist an elementary embedding \(f: V \rightarrow M\) with critical point \(\kappa \) such that \(j(\kappa )>\lambda \), \(V_\lambda \subset M\) and \(A\cap V_\lambda =j(A)\cap V_\lambda \).
Definition 5.8
A measurable cardinal is an uncountable cardinal \(\kappa \) such that there exists a \(\kappa \)complete nonprincipal ultrafilter on \(\kappa \).
Definition 5.9
A (strongly) inaccessible cardinal \(\kappa \) is such that \(\kappa \) is uncountable, regular and for every \(\lambda <\kappa \), \(2^\lambda <\kappa \).
Appendix B: Definitions and results
This appendix collects technical definitions and results left out in the main body of this article.
Results
The von Neumann hierarchy is formally defined as follows, where \(V\) is the universe of sets and the \(V_\alpha \) are levels within this universe.
Definition 6.1
\(V=\bigcup _{\alpha \in ON} V_\alpha \), where

\(V_0=\emptyset \)

\(V_{\alpha +1}=\mathcal {P}(V_\alpha )\)

\(V_\alpha =\bigcup \{V_\beta \mid \beta <\alpha \}\) where \(\alpha \) is a limit ordinal
The nonstationary ideal on \(\omega _1\), \({\mathcal {I}}_{NS}\), is defined as follows:
Definition 6.2
\({\mathcal {I}}_{NS}\) is the \(\sigma \)ideal of all sets \(A\subseteq \omega _1\) such that \(\omega _1\setminus A\) contains a closed unbounded set. A set \(S\subseteq \omega _1\) is stationary if for each closed unbounded set \(C\subseteq \omega _1\), \(S\cap C \ne \emptyset \). A set \(S\subseteq \omega _1\) is costationary if the complement of \(S\) is stationary (Woodin 2001b).
Universally baire sets
Definition 6.3
A set \(A\) in a compact Hausdorff space \(\Omega \) has the property of Baire if there is an open set \(O\subseteq \Omega \) such that the symmetric difference \(O\triangle A\) is meager. A meager set is a union of countably many nowhere dense sets. A nowhere dense set is a set whose closure has empty interior.
Definition 6.4
A set \(A\subseteq {\mathbb {R}}^n\) is called universally Baire if for every continuous function
where \(\Omega \) is a compact Hausdorff space, the preimage of \(A\) by \(F\) has the property of Baire.
Lemma 6.5

Every Borel set is universally Baire.

The universally Baire sets form a \(\sigma \)algebra, which is closed under preimages of Borel functions.

The universally Baire sets are Lebesgue measurable.
Definition 6.6
(Wadge Hierarchy) \(A<_W B\) iff \(A=f^{1}(B)\) for some continuous \(f: ^\omega \omega \rightarrow ^\omega \omega \) (Kanamori 2009).
In the presents of the Axiom of Determinacy, the Wadge hierarchy is a well order of the universally Baire sets, as Martin has shown in an unpublished article in 1973 called ‘The Wadge Degrees are well ordered’. In the absence of this axiom, the following theorem holds. Hereby is \({\mathbb {K}}\) the Cantor set and a set \(A\subseteq {\mathbb {K}}\) is said to be strongly reducible to \(B\subseteq {\mathbb {K}}\) if there is a continuous function \(g: {\mathbb {K}} \rightarrow {\mathbb {K}}\) such that \(A=f^{1}(B)\) and for all \(x,y\in {\mathbb {K}}\), \(\left f(x)f(y) \right \le (1/2)\left xy \right \).
Theorem 6.7
Suppose that \((A_k: k\in {\mathbb {N}})\) is a sequence of subsets of \({\mathbb {K}}\) such that for all \(k\in {\mathbb {N}}\) both \(A_{k+1}\) and \({\mathbb {K}}\setminus A_{k+1}\) are strongly reducible to \(A_k\). Then there exists a continuous function \(g: {\mathbb {K}} \rightarrow {\mathbb {K}}\) such that \(g^{1}(A_1)\) does not have the property of Baire (Woodin 2001b).
Because the universally Baire sets are closed under preimages of Borel functions, the above theorem shows wellfoundedness of \(<_W\) on the universally Baire sets.
Projective determinacy
As mentioned in the article, too much would have to be said to give an argument as to why the Axiom of Projective Determinacy, \(PD\), should be accepted. I will here only give a very brief sample case as presented in Woodin (2001a): the Banach–Tarski Paradox.
Given the unit sphere in a threedimensional space, there is a finite partition of the sphere into pieces which, after moving them around without changing their size, can be put together again to obtain two copies of the unit sphere. This is the Banach–Tarski Paradox. As is well known, it is an implication of the Axiom of Choice, \(AC\). Hence, if one wants to keep \(AC\), one may only hope to put constrains on the type of partitions in which the sphere may be divided. \(PD\) implies that these pieces may not be projective sets.
The projective sets are generalisations of the Borel sets.
Definition 6.8
(Luzin) A set \(X\subseteq {\mathbb {R}}^n\) is a projective set if for some integer \(k\) it can be generated from a closed subset of \({\mathbb {R}}^{n+k}\) in finitely many steps, applying the basic operations of taking projections and complements (Woodin 2001a).
The sets encountered in everyday mathematics are all projective sets. This may instil the idea that the projective sets are somehow ‘reasonable’. Hence, with \(PD\), any partition of the unit sphere into reasonable pieces will not allow for the paradox. This is seen by some as argument for the truth of \(PD\). See Woodin (2001a) for further details on \(PD\) and the Banach–Tarski–Paradox.
As mentioned in the above, \(ZFC+PD\) is not a forcing complete theory:
Theorem 6.9
\(PD\) is not forcing stable, i.e. there is a forcing which destroys \(PD\).
Proof
(sketch) Let \(A\) be a set of ordinals, which codes all of \({\mathbb {R}}\). If \(A^\sharp \) would exist in \(L[A]\), then there would be a Reinhardt cardinal in \(L[A]\). This contradicts \(AC\). Hence, \(A^\sharp \) does not exist in \(L[A]\). Now force with \(Col(\omega , A)\) to collapse \(A\) onto \(\omega \). Since \(A^\sharp \) is effectively a definable subset of \(A\) and \(Col\)forcing is homogeneous, \(A^\sharp \) does not exist in the forcing extension either. Since \(A\) is countable in the extension, \(A\) is effectively a real in the extension. By “\(\Pi ^1_1(x)\) determinacy iff \(x^\sharp \) exists” (proven by Harrington 1978) it follows that \(\Pi ^1_1(x)\) determinacy, and hence \(PD\), fails in the extension.^{Footnote 30} \(\square \)
Definitions
The following two definitions formally define what I have called ‘defining \(L\) by using the information coded in \(E\)’ and ‘adding \(A\) to \(L\)’, respectively.
Definition 6.10
Let \(E\) be a set. Then,

1.
\(L_0[E]=\emptyset \)

2.
\(L_{\alpha +1}=\fancyscript{P}_{Def}(Z)\), where
$$\begin{aligned} Z=L_\alpha [E]\cup \{E\cap L_\alpha [E]\} \end{aligned}$$and \(\fancyscript{P}_{Def}(Z)\) refers to the definable powerset of \(Z\).

3.
\(L_\alpha =\bigcup \{L_\beta \mid \beta ,\alpha \}\) for \(\alpha \) limit ordinal.
and \(L[E]\) is the class of all sets \(a\) such that \(a\in L_\alpha [E]\) for some \(\alpha \).
Definition 6.11
Suppose that \(A\) is a transitive set. Then,

1.
\(L_0(A)= A\)

2.
(Successor Case) \(L_{\alpha +1}(A)=\fancyscript{P}_{Def}(L_\alpha (A))\)

3.
(Limit Case) \(L_\alpha (A)= \bigcup \{L_\beta (A)\mid \beta <\alpha \}\)
and \(L(A)\) is the class of all sets \(a\) such that \(a\in L_\alpha (A)\) for some \(\alpha \).
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Rittberg, C.J. How Woodin changed his mind: new thoughts on the Continuum Hypothesis. Arch. Hist. Exact Sci. 69, 125–151 (2015). https://doi.org/10.1007/s0040701401428
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DOI: https://doi.org/10.1007/s0040701401428
Keywords
 Canonical Theory
 Force Extension
 Large Cardinal
 Measurable Cardinal
 Proper Class