How Woodin changed his mind: new thoughts on the Continuum Hypothesis

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.

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 truth-preserving 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 identity-embedding is the trivial elementary embedding and in all the definitions that follow $j$ is non-trivial.

Definition 5.1

A Reinhardt cardinal is a $\kappa =crit(j)$ for some elementary embedding $j: V \rightarrow V$.

Definition 5.2

An n-huge 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 non-principal ultra-filter 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 non-stationary 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 co-stationary 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

$$\begin{aligned} F: \Omega \rightarrow {\mathbb {R}}^n, \end{aligned}$$

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 $x,y\in {\mathbb {K}}$, $\left| f(x)-f(y) \right| \le (1/2)\left| x-y \right|$.

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 well-foundedness 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 three-dimensional 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.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 $\alpha$.