**Woodin cardinals** (named after W. Hugh Woodin) are a generalisation 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. Their exact definition has several equivalent but different characterisations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed. # Definitions An [[Inaccessible]] cardinal $\delta$ is *Woodin* if for every function $f:\delta\to\delta$ there exists $\kappa<\delta$ such that $f''\kappa\subseteq\kappa$ and a nontrivial [[Elementary embedding|elementary embedding]] $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)}\subseteq M$. ## Equivalent definitions Recall the following: ![[Strong#$A$-strongness]] - For an infinite ordinal $\delta$, a set $X\subseteq\delta$ is called *Woodin in $\delta$* if for every function $f:\delta\to\delta$, there is an ordinal $\kappa\in X$ with $f''\kappa\subseteq\kappa$ ($\kappa$ is closed under $f$) and there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)}\subseteq M$. Then for an inaccessible $\delta$, the following are equivalent to $\delta$ being Woodin. - For any set $A\subseteq V_\delta$, there exists a $\kappa<\delta$ that is ${<}\delta$-$A$-strong. I.e. $\forall A \subseteq V_{\delta}\exists\kappa<\delta\forall\gamma<\delta$ there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma}\subseteq M$ and $A\cap V_{\kappa+\gamma} = j(A)\cap V_{\kappa+\gamma}$. - For any set $A\subseteq V_\delta$, the set $S=\{\kappa<\delta:\kappa$ is ${<}\delta$-$A$-strong$\}$ is [[Club sets and stationary sets|stationary]] in $\delta$. - The set $F=\{X\subseteq \delta:\delta\setminus X \text{ is not Woodin in }\delta\}$ is a proper [[Filter]], the *Woodin filter* over $\delta$. # Some properties 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 $S\in F$. {% cite Kanamori2009 %} This implies every Woodin cardinal is [[Mahlo]] and preceded 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 preceded 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 non-stationary filter 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. ## Woodin cardinals and determinacy *See also: [[Axiom of determinacy]], [[Projective#Projective determinacy]] Woodin cardinals are linked to different forms of the [[Axiom of determinacy]] {% cite Kanamori2009 Larson2013 Koellner2010 %}: - $\text{ZF+AD}$, $\text{ZFC+AD}^{L(\mathbb{R})}$, 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". {% cite Koellner2010 %} 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 <a href="http://en.wikipedia.org/wiki/second-order_arithmetic" class="extiw" title="wikipedia:second-order arithmetic">second-order arithmetic</a>. - $\text{Z}_3+\Delta^1_2$-determinacy is provably equiconsistent with $\text{NBG}$+"$\text{Ord}$ is Woodin" where $\text{NBG}$ is <a href="http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" class="extiw" title="wikipedia:Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel set theory</a> and $\text{Z}_3$ is third-order arithmetic. ## Role in $\Omega$-logic ## Stationary tower forcing See [[Stationary tower forcing]]. # Woodin for strong compactness (from {% cite Dimopoulos2019 %} 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 lt;δ$-[](Strongly%20compact.md) for $A$. This definition is obviously analogous to one of the characterisations of Woodin and *Woodin-for-supercompactness* (Perlmutter proved that {% cite Perlmutter2010 %} it is equivalent to [Vopěnkaness](Vopenka.md "Vopenka")) cardinals. Results: - Woodin for strong compactness cardinal $δ$ is an [inaccessible](Inaccessible.md "Inaccessible") limits of lt;δ$-strongly compact cardinals. - $κ$ is Woodin and there are unboundedly many lt;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).