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Woodin cardinals (named after W. Hugh Woodin) are a generalization of the notion of strong cardinals (they give a model with many strengthened strong cardinals although they are usually not strong themselves) 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 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 [1]:

• 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 $S\in F$. [1] 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 (it is not $\Pi^1_1$-indescribable because of the extender definition of $\gamma$-$A$-strongness[2]).

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 $L(\mathbb{R})$ (relativized constructible universe; 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, while Woodins are usually not. However, just like Woodins, Shelah cardinals are weaker consistency-wise than superstrong cardinals.

If $κ$ is Shelah, then it is Woodin cardinal and there are $κ$ Woodin cardinals below it. If $κ$ is superstrong, then it is Shelah and there are $κ$ Shelah cardinals below it.[3]

Witnessing number

The witnessing number $wt(κ)$ of a Shelah cardinal $κ$ is the least cardinal $λ$ such that for any function $f : κ → κ$, there is an extender $E ∈ V_λ$ witnessing the Shelahness of $κ$ with respect to $f$.[3]

For every Shelah cardinal $κ$:[3]

• $κ$ is $ξ$-strong for all $ξ < wt(κ)$.
• Measurable Woodin cardinals are unbounded in $wt(κ)$.
• The following are equivalent:
• Witnessing numbers of Shelah cardinals less than $κ$ are unbounded in $κ$.
• Witnessing numbers of Shelah cardinals less than $wt(κ)$ are unbounded in $wt(κ)$.
• There is a Shelah cardinal $λ$ such that $κ < λ < wt(λ) < wt(κ)$.

Other properties

Small forcing preserves Shelah cardinals in both upward and downward directions.[4]

Shelah-for-supercompactness

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.

The least Shelah-for-supercompactness is not $\Sigma_2$-reflecting, so it is not supercompact and not even strong.[5]

Virtually Shelah for supercompactness

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 $f ∈ \mathrm{ran}(j)$.[6]

If $κ$ is virtually Shelah for supercompactness, then $V_κ$ is a model of proper class many virtually $C^{(n)}$-extendible cardinals for every $n < ω$.[6]

If $κ$ is 2-iterable, then $V_κ$ is a model of proper class many virtually Shelah for supercompactness cardinals.[6]

Woodin for strong compactness

(from [7] 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 [5] 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).

Hyper-Woodin

Definitions:[2]

• $U$ witnesses that $\delta$ is a hyper-Woodin cardinal iff $U$ is a normal measure on $\delta$ and for every set $A$, $\{\kappa < \delta | κ$ is $<\delta$-$A$-strong$\} \in U$.
• $\delta$ is weakly hyper-Woodin iff for every set $A$, there is a normal measure $U$ on $\delta$ such that $\{\kappa < \delta | κ$ is $<\delta$-$A$-strong$\} \in U$. (The difference is that here $U$ can depend on $A$.)

Results:[2]

• Superstrongness is consistency-wise stronger than hyper-Woodinness.
• If $\delta$ is hyper-Woodin, then $\delta$ is not the least measurable Woodin cardinal.
• If $\delta$ is a hyper-Woodin cardinal, then $\delta$ is Shelah in $N$, where $j : V \to N$ is the ultrapower map corresponding to $U$ that witnesses that $\delta$ is a hyper-Woodin cardinal..
• The least Shelah cardinal is strictly less than the least hyper-Woodin cardinal.
• If $\delta$ is a Shelah cardinal, then $\delta$ is a weakly hyper-Woodin cardinal.
• The least weakly hyper-Woodin cardinal is strictly less than the least Shelah cardinal.

Woodin cardinals and determinacy

Woodin cardinals are linked to different forms of the axiom of determinacy [1][8][9]:

• $\text{ZF+AD}$, $\text{ZFC+AD}^{L(\mathbb{R})}$, 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". [9] 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.

References

1. Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www   bibtex
2. Schimmerling, Ernest. Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model. Proc Amer Math Soc 130(11):3385-3391, 2002. DOI   bibtex
3. Golshani, Mohammad. An Easton like theorem in the presence of Shelah cardinals. M Arch Math Logic 56(3-4):273-287, May, 2017. www   DOI   bibtex
4. Daghighi, Ali Sadegh and Pourmahdian, Massoud. On Some Properties of Shelah Cardinals. Bull Iran Math Soc 44(5):1117-1124, October, 2018. www   DOI   bibtex
5. Perlmutter, Norman. The large cardinals between supercompact and almost-huge. , 2010. www   arχiv   bibtex
6. Gitman, Victoria and Shindler, Ralf. Virtual large cardinals. www   bibtex
7. Dimopoulos, Stamatis. Woodin for strong compactness cardinals. The Journal of Symbolic Logic 84(1):301–319, 2019. arχiv   DOI   bibtex
8. Larson, Paul B. A brief history of determinacy. , 2013. www   bibtex
9. Koellner, Peter and Woodin, W Hugh. Chapter 23: Large cardinals from Determinacy. Handbook of Set Theory , 2010. www   bibtex
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