Determinacy from strong compactness of ω₁

Abstract

In the absence of the Axiom of Choice, the “small” cardinal ${\omega }_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that ${\omega }_1$ is X-strongly compact (where X is any set) if there is a fine, countably complete measure on ${\wp }_{{\omega }_1}(X)$. Working in $\mathsf{ZF}+\mathsf{DC}$, we prove that the $\wp ({\omega }_1)$-strong compactness and $\wp (\mathbb{R})$-strong compactness of ${\omega }_1$ are equiconsistent with $\mathsf{AD}$ and ${\mathsf{AD}}_{\mathbb{R}}+\mathsf{DC}$ respectively, where $\mathsf{AD}$ denotes the Axiom of Determinacy and ${\mathsf{AD}}_{\mathbb{R}}$ denotes the Axiom of Real Determinacy. The $\wp (\mathbb{R})$-supercompactness of ${\omega }_1$ is shown to be slightly stronger than ${\mathsf{AD}}_{\mathbb{R}}+\mathsf{DC}$, but its consistency strength is not computed precisely. An equiconsistency result at the level of ${\mathsf{AD}}_{\mathbb{R}}$ without $\mathsf{DC}$ is also obtained.

Introduction

We assume $\mathsf{ZF}+\mathsf{DC}$ as our background theory unless otherwise stated. (However, we will sometimes weaken our choice principle to a fragment of $\mathsf{DC}$.) In this setting, it is possible for ${\omega }_1$ to exhibit “large cardinal” properties such as strong compactness. The appropriate definition of strong compactness is made in terms of measures (ultrafilters) on sets of the form ${\wp }_{{\omega }_1}(X)$.

Definition 1.1

Let X be an uncountable set. A measure μ on ${\wp }_{{\omega }_1}(X)$ is countably complete if it is closed under countable intersections and fine if it contains the set $\{\sigma \in {\wp }_{{\omega }_1}(X):x\in \sigma \}$ for all $x\in X$. We say that ${\omega }_1$ is X-strongly compact if there is a countably complete fine measure on ${\wp }_{{\omega }_1}(X)$.

For uncountable sets X and Y, we will often use the elementary fact that if ${\omega }_1$ is X-strongly compact and there is a surjection from X to Y, then ${\omega }_1$ is Y-strongly compact as witnessed by a push-forward measure.

In the absence of $\mathsf{AC}$, it may become necessary to consider degrees X of strong compactness that are not wellordered. The first and most important example is $X=\mathbb{R}$. The theory $\mathsf{ZFC}+$“there is a measurable cardinal” is equiconsistent with the theory $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is $\mathbb{R}$-strongly compact.” (For a proof of the forward direction, see Trang [19]. The reverse direction is proved by noting that ${\omega }_1$ is ${\omega }_1$-strongly compact, hence measurable, and considering an inner model $L(\mu )$ where μ is a measure on ${\omega }_1$.)

Another way to obtain $\mathbb{R}$-strong compactness of ${\omega }_1$ that is more relevant to this paper is by the Axiom of Determinacy. If $\mathsf{AD}$ holds then by Martin's cone theorem, for every set $A\in {\wp }_{{\omega }_1}(\mathbb{R})$ the property $\{x\in \mathbb{R}:x{\le }_{\text{T}}d\}\in A$ either holds for a cone of Turing degrees d or fails for a cone of Turing degrees d, giving a countably complete fine measure on ${\wp }_{{\omega }_1}(\mathbb{R})$.

Besides $\mathbb{R}$, another relevant degree of strong compactness is the cardinal Θ, which is defined as the least ordinal that is not a surjective image of $\mathbb{R}$. In other words, Θ is the successor of the continuum in the sense of surjections. If the continuum can be wellordered then this is the same as the successor in the sense of injections (that is, ${\mathfrak{c}}^{+}$.) However in general it can be much larger. For example, if $\mathsf{AD}$ holds then Θ is strongly inaccessible by Moschovakis's coding lemma, but on the other hand there is no injection from ${\omega }_1$ into $\mathbb{R}$.

If ${\omega }_1$ is $\mathbb{R}$-strongly compact, then pushing forward a measure witnessing this by surjections, we see that ${\omega }_1$ is λ-strongly compact for every uncountable cardinal $\lambda <\mathrm{\Theta }$. In general all we can say is $\mathrm{\Theta }\ge {\omega }_2$ and so this does not give anything beyond measurability of ${\omega }_1$. However, it does suggest two marginal strengthenings of the hypothesis “${\omega }_1$ is $\mathbb{R}$-strongly compact” with the potential to increase the consistency strength beyond measurability. Namely, we may add the hypothesis “${\omega }_1$ is ${\omega }_2$-strongly compact” or the hypothesis “${\omega }_1$ is Θ-strongly compact.” We will consider both strengthenings and obtain equiconsistency results in both cases.

In order to state and obtain sharper results, we first recall some combinatorial consequences of strong compactness. Let λ be an infinite cardinal and let $\overrightarrow{C}=(C_{\alpha }:\alpha \in \lim (\lambda ))$ be a sequence such that each set $C_{\alpha }$ is a club subset of α. The sequence $\overrightarrow{C}$ is coherent if for all $\beta \in \lim (\lambda )$ and all $\alpha \in \lim (C_{\beta })$ we have $C_{\alpha }=C_{\beta }\cap \alpha$. A thread for a coherent sequence $\overrightarrow{C}$ is a club subset $D\subset \lambda$ such that for all $\alpha \in \lim (D)$ we have $C_{\alpha }=D\cap \alpha$. An infinite cardinal λ is called threadable if every coherent sequence of length λ has a thread. Threadability of λ is also known as $\neg \square (\lambda )$.

The following result is a well-known consequence of the “discontinuous ultrapower” characterization of strong compactness. However, without $\mathsf{AC}$ Łoś's theorem may fail for ultrapowers of V, so we must verify that the argument can be done using ultrapowers of appropriate inner models instead.

Lemma 1.2

Assume $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is λ-strongly compact” where λ is a cardinal of uncountable cofinality. Then λ is threadable.

Proof

Let $\overrightarrow{C}=(C_{\alpha }:\alpha \in \lim (\lambda ))$ be a coherent sequence such that each set $C_{\alpha }$ is a club in α. Consider the $\mathsf{ZFC}$ model $L[\{(\alpha ,\beta ):\alpha \in C_{\beta }\}]$, which we abbreviate as $L[\overrightarrow{C}]$. Let μ be a countably complete fine measure on ${\wp }_{{\omega }_1}(\lambda )$ and let $j:L[\overrightarrow{C}]\to \mathrm{Ult}(L[\overrightarrow{C}],\mu )$ be the corresponding ultrapower map, where the ultrapower is defined using all functions ${\wp }_{{\omega }_1}(\lambda )\to L[\overrightarrow{C}]$ in V. The ultrapower is wellfounded by countable completeness and $\mathsf{DC}$, so it has the form $L[j(\overrightarrow{C})]$. Note that j is discontinuous at λ: for any ordinal $\alpha <\lambda$, we have $j(\alpha )\le {[\sigma \mapsto \sup \sigma ]}_{\mu }<j(\lambda )$ where the first inequality holds because μ is fine and the second inequality holds because λ has uncountable cofinality.

Now the argument continues as usual. We define the ordinal $\gamma =\sup j[\lambda ]$ and note that $\gamma <j(\lambda )$ and that $j[\lambda ]$ is an ω-club in γ. Therefore the set $j[\lambda ]\cap \lim (j{(\overrightarrow{C})}_{\gamma })$ is unbounded in γ, so its preimage $S=j^{-1}[\lim (j{(\overrightarrow{C})}_{\gamma })]$ is unbounded in λ. Note that the club $C_{\alpha }$ is an initial segment of $C_{\beta }$ whenever $\alpha ,\beta \in S$ and $\alpha <\beta$; this is easy to check using the elementarity of j and the coherence of $j(\overrightarrow{C})$. Therefore the union of clubs ${\bigcup }_{\alpha \in S}{C}_{\alpha }$ threads the sequence $\overrightarrow{C}$. □

If $\lambda <\mathrm{\Theta }$ then ${\mathsf{DC}}_{\mathbb{R}}$ suffices in place of $\mathsf{DC}$:

Lemma 1.3

Assume $\mathsf{ZF}+{\mathsf{DC}}_{\mathbb{R}}+$“${\omega }_1$ is $\mathbb{R}$-strongly compact.” Let $\lambda <\mathit{\Theta }$ be a cardinal of uncountable cofinality. Then λ is threadable.

Proof

Let $\overrightarrow{C}$ be a coherent sequence of length λ. First, note that we may pass to an inner model containing $\overrightarrow{C}$ where $\mathsf{DC}$ holds in addition to our other hypotheses. Namely, let $f:\mathbb{R}\to \lambda$ be a surjection, let μ be a fine, countably complete measure on ${\wp }_{{\omega }_1}(\mathbb{R})$, let $C=\{(\alpha ,\beta ):\alpha \in C_{\beta }\}$, and consider the model $M=L(\mathbb{R})[f,\mu ,C]$, where the square brackets indicate that we are constructing from f, μ and C as predicates. (In the case of μ, this distinction is important: we are not putting all elements of μ into the model.)

It can be easily verified that all of our hypotheses are downward absolute to the model M, and that our desired conclusion that $\overrightarrow{C}$ has a thread is upward absolute from M to V. In the model M every set is a surjective image of $\mathbb{R}\times \alpha$ for some ordinal α, so $\mathsf{DC}$ follows from ${\mathsf{DC}}_{\mathbb{R}}$ by a standard argument. Moreover, ${\omega }_1$ is λ-strongly compact in M by pushing forward the measure μ (restricted to M) by the surjection f, so the desired result follows from Lemma 1.2. □

A further combinatorial consequence of strong compactness of ${\omega }_1$ is the failure of Jensen's square principle ${\square }_{{\omega }_1}$. In fact $\neg {\square }_{{\omega }_1}$ follows from the assumption that ${\omega }_2$ is threadable or singular (note that successor cardinals may be singular in the absence of $\mathsf{AC}$.)

Lemma 1.4

Assume $\mathsf{ZF}$. If ${\omega }_2$ is singular or threadable, then $\neg {\square }_{{\omega }_1}$.

Proof

Suppose toward a contradiction that ${\omega }_2$ is singular or threadable and we have a ${\square }_{{\omega }_1}$-sequence $(C_{\alpha }:\alpha \in \lim ({\omega }_2))$. If ${\omega }_2$ is singular, we do not need coherence of the sequence to reach a contradiction. Take any cofinal set $C_{{\omega }_2}$ in ${\omega }_2$ of order type $\le {\omega }_1$ and recursively define a sequence of functions $(f_{\alpha }:\alpha \in [{\omega }_1,{\omega }_2])$ such that each function $f_{\alpha }$ is a surjection from ${\omega }_1$ onto α, using our small cofinal sets $C_{\alpha }$ at limit stages. Then the function $f_{{\omega }_2}$ is a surjection from ${\omega }_1$ onto ${\omega }_2$, a contradiction. On the other hand, if ${\omega }_2$ is regular and threadable, take a thread $C_{{\omega }_2}$ through the square sequence. Then by the usual argument the order type of $C_{{\omega }_2}$ is at most ${\omega }_1+\omega$, contradicting the regularity of ${\omega }_2$. □

Now we can state our equiconsistency results and prove their easier directions.

Theorem 1.5

The following theories are equiconsistent:

• $\mathsf{ZF}+\mathsf{DC}+\mathsf{AD}$.

• $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is $\wp ({\omega }_1)$-strongly compact.”

• $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is $\mathbb{R}$-strongly compact and ${\omega }_2$-strongly compact.”

• $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is $\mathbb{R}$-strongly compact and $\neg {\square }_{{\omega }_1}$.”

Proof

(1) ⇒ (2): Under $\mathsf{AD}$, Martin's cone theorem implies that ${\omega }_1$ is $\mathbb{R}$-strongly compact. There is a surjection from $\mathbb{R}$ onto $\wp ({\omega }_1)$ by Moschovakis's coding lemma, so ${\omega }_1$ is $\wp ({\omega }_1)$-strongly compact as well.

(2) ⇒ (3): This follows from the existence of surjections from $\wp ({\omega }_1)$ onto $\mathbb{R}$ and ${\omega }_2$.

(3) ⇒ (4): This follows from Lemma 1.2, Lemma 1.4.

Con (4)⇒Con (1): In Sections 3 and 4, we will show that statement (4) implies ${\mathsf{AD}}^{L(\mathbb{R})}$. □

Moving up the consistency strength hierarchy, the next natural target for an equiconsistency result is the theory $\mathsf{ZF}+{\mathsf{AD}}_{\mathbb{R}}$. Here ${\mathsf{AD}}_{\mathbb{R}}$ denotes the Axiom of Determinacy for real games, which has higher consistency strength than $\mathsf{AD}$ and cannot hold in $L(\mathbb{R})$. To get a model of ${\mathsf{AD}}_{\mathbb{R}}$ we will need to augment our strong compactness hypothesis somehow, for example with a hypothesis on Θ or $\wp (\mathbb{R})$. For any set X, we write ${\mathsf{DC}}_X$ for the fragment of $\mathsf{DC}$ that allows us to choose ω-sequences of subsets of X.

Theorem 1.6

The following theories are equiconsistent:

• $\mathsf{ZF}+{\mathsf{AD}}_{\mathbb{R}}$.

• $\mathsf{ZF}+{\mathsf{DC}}_{\wp ({\omega }_1)}+$“${\omega }_1$ is $\mathbb{R}$-strongly compact and Θ is singular.”

Proof

Con (1)⇒Con (2): By Solovay [11], if $\mathsf{ZF}+{\mathsf{AD}}_{\mathbb{R}}$ is consistent then so is $\mathsf{ZF}+{\mathsf{AD}}_{\mathbb{R}}+$“Θ is singular.” (In particular Solovay showed that the cofinality of Θ can be countable, which implies the failure of $\mathsf{DC}$.) Under ${\mathsf{AD}}_{\mathbb{R}}$ we have that ${\omega }_1$ is $\mathbb{R}$-strongly compact by Martin's measure (this just follows from $\mathsf{AD}$) and we have ${\mathsf{DC}}_{\mathbb{R}}$ (this follows from uniformization for total relations on $\mathbb{R}$.) Moreover there is a surjection from $\mathbb{R}$ to $\wp ({\omega }_1)$ by the coding lemma, so ${\mathsf{DC}}_{\mathbb{R}}$ can be strengthened to ${\mathsf{DC}}_{\wp ({\omega }_1)}$.

Con (2)⇒Con (1): In Sections 5 and 6, we will show that if statement (2) holds, then statement (1) holds in an inner model of the form $L({\mathrm{\Omega }}^{⁎},\mathbb{R})$ where ${\mathrm{\Omega }}^{⁎}\subset \wp (\mathbb{R})$. Note that statement (2) implies that ${\omega }_2$ is either singular (if ${\omega }_2=\mathrm{\Theta }$) or threadable (if ${\omega }_2<\mathrm{\Theta }$, by Lemma 1.3) so in either case we have $\neg {\square }_{{\omega }_1}$ by Lemma 1.4. Therefore we can make some use of the argument for Con (4)⇒Con (1) of Theorem 1.5 here, once we check that ${\mathsf{DC}}_{\wp ({\omega }_1)}$ suffices in place of $\mathsf{DC}$ for this argument. □

Finally, we will obtain an equiconsistency result at the level of $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$. Note that this theory has strictly higher consistency strength than $\mathsf{ZF}+{\mathsf{AD}}_{\mathbb{R}}$. (By contrast, $\mathsf{ZF}+\mathsf{DC}+\mathsf{AD}$ and $\mathsf{ZF}+\mathsf{AD}$ are equiconsistent by a theorem of Kechris.)

Theorem 1.7

The following theories are equiconsistent:

• $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$.

• $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is $\wp (\mathbb{R})$-strongly compact.”

• $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is $\mathbb{R}$-strongly compact and Θ-strongly compact.”

• $\mathsf{ZF}+\mathsf{DC}+$“${\omega }_1$ is $\mathbb{R}$-strongly compact and Θ is singular.”

Proof

Con (1)⇒Con (2): By Solovay [11], under $\mathsf{ZF}+{\mathsf{AD}}_{\mathbb{R}}$ we have $\mathsf{DC}$ if and only if Θ has uncountable cofinality, and in a minimal model of $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$ we have that Θ is singular of cofinality ${\omega }_1$. Assume that we are in such a minimal model of $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$ and take a cofinal increasing function $\pi :{\omega }_1\to \mathrm{\Theta }$.

We can express $\wp (\mathbb{R})$ as an increasing union ${\bigcup }_{\alpha <{\omega }_1}{\mathrm{\Gamma }}_{\alpha }$ where the pointclass ${\mathrm{\Gamma }}_{\alpha }$ consists of all sets of reals of Wadge rank at most $\pi (\alpha )$. For each $\alpha <{\omega }_1$ there is a surjection from $\mathbb{R}$ onto ${\mathrm{\Gamma }}_{\alpha }$, so ${\omega }_1$ is ${\mathrm{\Gamma }}_{\alpha }$-strongly compact. Moreover, ${\mathsf{AD}}_{\mathbb{R}}$ implies that there is a uniform way to choose, for each $\alpha <{\omega }_1$, a countably complete fine measure ${\mu }_{\alpha }$ on ${\wp }_{{\omega }_1}({\mathrm{\Gamma }}_{\alpha })$ witnessing this fact (namely the unique normal fine measure; see Woodin [23, Theorem 4].)

Using a countably complete nonprincipal measure ν on ${\omega }_1$ (which exists because ${\omega }_1$ is ${\omega }_1$-strongly compact) we can assemble these measures ${\mu }_{\alpha }$ into a countably complete fine measure ${\mu }^{⁎}$ on ${\wp }_{{\omega }_1}(\wp (\mathbb{R}))$ as follows: for $A\subseteq {\wp }_{{\omega }_1}(\wp (\mathbb{R}))$, we say $A\in {\mu }^{⁎}\iff {\forall }_{\nu }^{⁎}\alpha A\cap {\wp }_{{\omega }_1}({\mathrm{\Gamma }}_{\alpha })\in {\mu }_{\alpha }$. It's easy to verify that ${\mu }^{⁎}$ is countably complete because ν and the ${\mu }_{\alpha }$'s are countably complete. Likewise, it's easy to verify that ${\mu }^{⁎}$ is fine because ν is uniform and the ${\mu }_{\alpha }$'s are fine. Therefore the measure ${\mu }^{⁎}$ witnesses that ${\omega }_1$ is $\wp (\mathbb{R})$-strongly compact, so statement (2) holds (in our minimal model of $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$.)

(2) ⇒ (3): This follows from the existence of surjections from $\wp (\mathbb{R})$ onto $\mathbb{R}$ and Θ.

Con (1)⇒Con (4): This follows by the aforementioned result of Solovay that in a minimal model of $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$ the cardinal Θ is singular of cofinality ${\omega }_1$ (and of course ${\omega }_1$ is $\mathbb{R}$-strongly compact by Martin's measure).

Con (3)∨Con (4)⇒Con (1): We will show in Sections 5 and 6 that if either statement (3) or statement (4) holds, then statement (1) holds in an inner model of the form $L({\mathrm{\Omega }}^{⁎},\mathbb{R})$ where ${\mathrm{\Omega }}^{⁎}\subset \wp (\mathbb{R})$. The proof of Con (4)⇒Con (1) is similar to the proof of Con (2)⇒Con (1) in Theorem 1.7, although one should note that the inner model $L({\mathrm{\Omega }}^{⁎},\mathbb{R})$ does not simply absorb $\mathsf{DC}$ from V; a bit more argument is required. □

The authors would like to thank the referee for a careful reading of this article and for pointing out several minor errors. The first author would like to thank the NSF for its generous support through grant DMS-1849295.