Woodin Cardinal

Woodin cardinals 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. 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 Shelah 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 $V_{j(f)(\kappa)}$.

Shelah cardinals are above strong but below superstrong in the large cardinal/consistency hierarchy.

It turns out that Shelah cardinals have many large cardinals below them that suffice to establish the regularity properties, and as a result have mostly faded from view in the large cardinal research literature.

Woodin cardinals

Woodin cardinals are a refinement of Shelah cardinals. The primary difference is the requirement of a closure condition on the functions $f:\kappa\rightarrow\kappa$ and associated embeddings. Woodin cardinals are not themselves the critical points of any of their associated embeddings and hence need not be measurable. They are, however, Mahlo cardinals (and hence also inaccessible) since the set of measurable cardinals below a Woodin cardinal must be stationary.

Elementary Embedding Characterization

A cardinal $\kappa$ is "Woodin" if, for every function $f:\kappa\to\kappa$ there is some $\gamma<\kappa$ such that $f$ is closed under $\gamma$, i.e. $\{f(\alpha)|\alpha\in\gamma\}\subseteq\gamma$ and there is an associated non-trivial elementary embedding $j:V\prec M$ with critical point $\gamma$ such that $V_{j(f)(\kappa)}\subseteq M$

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., $A\cap V_{\alpha+\gamma}=j_\gamma(A)\cap V_{\alpha+\gamma}$, have critical point $\alpha$, and whose target transitive class contains the initial segment $V_{\alpha+\gamma}$.

Analogue of Vopěnka's Principle

This material will be added later.

See also the Woodin for supercompactness cardinals, which are identical to the Vopěnka cardinals.

References

1. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
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