Woodin Cardinal

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Woodin cardinals are a generalization of the notion of strong cardinals and they have been used to calibrate the exact proof-theoretic strength of the Axiom of Determinacy. Woodin cardinals are weaker than superstrong cardinals in consistency strength and fail to be weakly compact in general, since they are not $\Pi_1^1$ indescribable. 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.

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 non-trivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point $<\lambda$. This 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.

A cardinal $\kappa$ is a Shelah cardinal if, for every function $f:\kappa\to\kappa$ there is a non-trivial elementary embedding $j:V\prec M$ with $M$ a transitive class, $\kappa$ the critical point of $j$ and $M$ contains the initial segment $V_{j(f)(\kappa)}$.

Elementary Embedding Characterization

Woodin cardinals are modelled after Shelah cardinals except for the addition of a closure condition on the functions $f:\kappa\rightarrow\kappa$.

A cardinal $\kappa$ is a Woodin cardinal if, for every function $f:\kappa\to\kappa$ there is some $\gamma<\kappa$ such that $f$ is closed under $\gamma$ and a non-trivial elementary embedding $j:V\prec M$ with critical point $\gamma$ and $M$ containing the initial segment $V_{(j(f))(\alpha)}$.

If $\kappa$ is Woodin then for any subset $A\subseteq V_\kappa$ some $\alpha <\kappa$ is $\gamma$-strong for every $\gamma <\kappa$. Intuitively this means that there are elementary embeddings $j_\gamma$ which preserve $A$ i.e., $A\cap V_{\alpha+\gamma}=j_\gamma(A)\cap V_{\alpha+\gamma}$ and critical point $\alpha$ whose target transitive class contains the initial segment $V_{\alpha+\gamma}$.

There is a hierarchy of Woodin-type cardinals

Analogue of Vopenka's Principle

Stationary Tower Forcing

Role in $\Omega$-Logic and the Resurrection Theorem

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