Woodin cardinal

• Definition and some properties
• Shelah cardinals
• Woodin for strong compactness
• Woodin cardinals and determinacy
• Role in $\Omega$-logic
• Stationary tower forcing

Woodin cardinals (named after W. Hugh Woodin) are a generalization of the notion of strong cardinals and have been used to calibrate the exact proof-theoretic strength of the axiom of determinacy. They can also be seen as weakenings of Shelah cardinals, defined below. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed.

Definition and some properties

We first introduce the concept of $\gamma$-strongness for $A$: an ordinal $\kappa$ is $\gamma$-strong for $A$ (or $\gamma$-$A$-strong) if there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma}\subseteq M$ and $A\cap V_{\kappa+\gamma} = j(A)\cap V_{\kappa+\gamma}$. Intuitively, $j$ preserves the part of $A$ that is in $V_{\kappa+\gamma}$. We say that a cardinal $\kappa$ is <$\delta$-$A$-strong if it is $\gamma$-$A$-strong for all $\gamma<\delta$.

We also introduce Woodin-ness in $\delta$: for an infinite ordinal $\delta$, a set $X\subseteq\delta$ is Woodin in $\delta$ if for every function $f:\delta\to\delta$, there is an ordinal $\kappa\in X$ with $\{f(\beta):\beta<\kappa\}\subseteq\kappa$ ($\kappa$ is closed under $f$), there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)}\subseteq M$.

An inaccessible cardinal $\delta$ is Woodin if any of the following (equivalent) characterizations holds (Kanamori, 2009):

• For any set $A\subseteq V_\delta$, there exists a $\kappa<\delta$ that is <$\delta$-$A$-strong.
• For any set $A\subseteq V_\delta$, the set $S=\{\kappa<\delta:\kappa$ is <$\delta$-$A$-strong$\}$ is stationary in $\delta$.
• The set $F=\{X\subseteq \delta:\delta\setminus X$ is not Woodin in $\delta$$\}$ is a proper filter, the Woodin filter over $\delta$.
• For every function $f:\delta\to\delta$ there exists $\kappa<\delta$ such that $\{f(\beta):\beta\in\kappa\}\subseteq\kappa$ (that is, $\kappa$ is closed under $f$) and there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)}\subseteq M$.

Let $\delta$ be Woodin, $F$ be the Woodin filter over $\delta$, and $S=\{\kappa<\delta:\kappa$ is <$\delta$-$A$-strong$\}$. Then $F$ is normal and $S\in F$. (Kanamori, 2009) This implies every Woodin cardinal is Mahlo and preceeded by a stationary set of measurable cardinals, in fact of <$\delta$-strong cardinals. However, the least Woodin cardinal is not weakly compact as it is not $\Pi^1_1$-indescribable.

Woodin cardinals are weaker consistency-wise then superstrong cardinals. In fact, every superstrong is preceeded by a stationary set of Woodin cardinals. On the other hand the existence of a Woodin is much stronger than the existence of a proper class of strong cardinals.

The existence of a Woodin cardinal implies the consistency of $\text{ZFC}$ + “the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated”. Huge cardinals were first invented to prove the consistency of the existence of a $\omega_2$-saturated $\sigma$-ideal on $\omega_1$, but turned out to be stronger than required, as a Woodin is enough.

Shelah cardinals

Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $L(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc…). In slightly more detail, Woodin had established that the axiom of determinacy (a hypothesis known to imply regularity properties for sets of reals) holds in $L(\mathbb{R})$ assuming the existence of a nontrivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point $<\lambda$. This axiom, a rank-into-rank axiom, is known to be very strong and its use was first weakened to that of the existence of a supercompact cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the regularity properties of sets of reals mentioned above.

Shelah cardinals were the first cardinals to be devised by Woodin and Shelah. A cardinal $\delta$ is Shelah if for every function $f:\delta\to\delta$ there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\delta$ such that $V_{j(f)(\delta)}\subseteq M$. Every Shelah is Woodin, but not every Woodin is Shelah: indeed, Shelah cardinals are always measurable and in fact strong, while Woodins are usually not. However, just like Woodins, Shelah cardinals are weaker consistency-wise than superstrong cardinals.

A related notion is Shelah-for-supercompactness, where the closure condition $V_{j(f)(\delta)}\subseteq M$ is replaced by $M^{j(f)(\delta)}\subseteq M$, a much stronger condition. The difference between Shelah and Shelah-for-supercompactness cardinals is essentially the same as the difference between strong and supercompact cardinals, or between superstrong and huge cardinals. Also, just like every Shelah is preceeded by a stationary set of strong cardinals, every Shelah-for-supercompactness cardinal is preceeded by a stationary set of supercompact cardinals.

Much weaker, consistent with $V=L$ variant: A cardinal $κ$ is virtually Shelah for supercompactness iff for every function $f : κ → κ$ there are $λ > κ$ and $\bar{λ}< κ$ such that in a set-forcing extension there is an elementary embedding $j : V_{\bar{λ}}→ V_{λ}$ with $j(\mathrm{crit}(j)) = κ$, $\bar{λ} ≥ f(\mathrm{crit}(j))$ and $f ∈ \mathrm{ran}(j)$. If $κ$ is virtually Shelah for supercompactness, then $V_κ$ is a model of proper class many virtually $C^{(n)}$-extendible cardinals for every $n < ω$ and if κ is 2-iterable, then $V_κ$ is a model of proper class many virtually Shelah for supercompactness cardinals.(Gitman & Shindler, n.d.)

Woodin for strong compactness

(from (Dimopoulos, 2019) unless otherwise noted)

A cardinal $δ$ is Woodin for strong compactness (or Woodinised strongly compact) iff for every $A ⊆ δ$ there is $κ < δ$ which is $<δ$-strongly compact for $A$.

This definition is obviously analogous to one of the characterisations of Woodin and Woodin-for-supercompactness (Perlmutter proved that (Perlmutter, 2010) it is equivalent to Vopěnkaness) cardinals.

Results:

• Woodin for strong compactness cardinal $δ$ is an inaccessible limits of $<δ$-strongly compact cardinals.
• $κ$ is Woodin and there are unboundedly many $<δ$-supercompact cardinals below $δ$, then $δ$ is Woodin for strong compactness.
• The existence of a Woodin for strong compactness cardinal is at least as strong as a proper class of strongly compact cardinals and at most as strong as a Woodin limit of supercompact cardinals (which lies below an extendible cardinal).

Woodin cardinals and determinacy

See also: axiom of determinacy, axiom of projective determinacy

Woodin cardinals are linked to different forms of the axiom of determinacy (Kanamori, 2009; Larson, 2013; Koellner & Woodin, 2010):

• $\text{ZF+AD}$, $\text{ZFC+AD}^{L(\mathbb{R})}$, ZFC+”the non-stationary ideal over $\omega_1$ is $\omega_1$-dense” and $\text{ZFC}$+”there exists infinitely many Woodin cardinals” are equiconsistent.
• Under $\text{ZF+AD}$, the model $\text{HOD}^{L(\mathbb{R})}$ satisfies $\text{ZFC}$+”$\Theta^{L(\mathbb{R})}$ is a Woodin cardinal”. (Koellner & Woodin, 2010) gives many generalizations of this result.
• If there exists infinitely many Woodin cardinals with a measurable above them all, then $\text{AD}^{L(\mathbb{R})}$. If there assumtion that there is a measurable above those Woodins is removed, one still has projective determinacy.
• In fact projective determinacy is equivalent to “for every $n<\omega$, there is a fine-structural, countably iterable inner model $M$ such that $M$ satisfies $\text{ZFC}$+”there exists $n$ Woodin cardinals”.
• For every $n$, if there exists $n$ Woodin cardinals with a measurable above them all, then all $\mathbf{\Sigma}^1_{n+1}$ sets are determined.
• $\mathbf{\Pi}^1_2$-determinacy is equivalent to “for every $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of $\text{ZFC}$ containing $x$.
• $\mathbf{\Delta}^1_2$-determinacy is equivalent to “for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M$ satisfies ZFC+”there is a Woodin cardinal”.
• $\text{ZFC}$ + lightface $\Delta^1_2$-determinacy implies that there many $x$ such that $\text{HOD}^{L[x]}$ satisfies $\text{ZFC}$+”$\omega_2^{L[x]}$ is a Woodin cardinal”.
• $\text{Z}_2+\Delta^1_2$-determinacy is conjectured to be equiconsistent with $\text{ZFC}$+”$\text{Ord}$ is Woodin”, where “$\text{Ord}$ is Woodin” is expressed as an axiom scheme and $\text{Z}_2$ is second-order arithmetic.
• $\text{Z}_3+\Delta^1_2$-determinacy is provably equiconsistent with $\text{NBG}$+”$\text{Ord}$ is Woodin” where $\text{NBG}$ is Von Neumann–Bernays–Gödel set theory and $\text{Z}_3$ is third-order arithmetic.

Role in $\Omega$-logic

Stationary tower forcing

References

1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
2. Gitman, V., & Shindler, R. Virtual large cardinals. https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf
3. Dimopoulos, S. (2019). Woodin for strong compactness cardinals. The Journal of Symbolic Logic, 84(1), 301–319. https://doi.org/10.1017/jsl.2018.67
4. Perlmutter, N. (2010). The large cardinals between supercompact and almost-huge. http://boolesrings.org/perlmutter/files/2013/07/HighJumpForJournal.pdf
5. Larson, P. B. (2013). A brief history of determinacy. http://www.users.miamioh.edu/larsonpb/determinacy_cabal.pdf
6. Koellner, P., & Woodin, W. H. (2010). Chapter 23: Large cardinals from Determinacy. Handbook of Set Theory. http://logic.harvard.edu/koellner/LCFD.pdf

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