Woodin cardinal

The hierarchy V_\alpha (known as the von Neumann hierarchy) is defined by transfinite recursion on \alpha: • V_0 = \varnothing, • V_{\alpha+1} = \mathcal P(V_\alpha), • V_\alpha = \bigcup_{\beta, when \alpha is a limit ordinal. For any ordinal \alpha, V_\alpha is a set. The union of the sets V_\alpha for all ordinals \alpha is no longer a set, but a proper class. Some of the sets V_\alpha have set-theoretic properties, for example when \kappa is an inaccessible cardinal, V_\kappa satisfies second-order ZFC ("satisfies" here means the notion of satisfaction from first-order logic). For a transitive class M, a function j:V\to M is said to be an elementary embedding if for any formula \phi with free variables x_1,\ldots,x_n in the language of set theory, it is the case that V\vDash\phi(x_1,\ldots,x_n) iff M\vDash\phi(j(x_1),\ldots,j(x_n)), where \vDash is first-order logic's notion of satisfaction as before. An elementary embedding j is called nontrivial if it is not the identity. If j:V\to M is a nontrivial elementary embedding, there exists an ordinal \kappa such that j(\kappa)\neq\kappa, and the least such \kappa is called the critical point of j. Many large cardinal properties can be phrased in terms of elementary embeddings. For an ordinal \beta, a cardinal \kappa is said to be \beta-strong if a transitive class M can be found such that there is a nontrivial elementary embedding j:V\to M whose critical point is \kappa, and in addition V_\beta\subseteq M. A strengthening of the notion of \beta-strong cardinal is the notion of A-strongness of a cardinal \kappa in a greater cardinal \delta: if \kappa and \delta are cardinals with \kappa, and A is a subset of V_\delta, then \kappa is said to be A-strong in \delta if for all \beta, there is a nontrivial elementary embedding j:V\to M witnessing that \kappa is \beta-strong, and in addition j(A)\cap V_\beta = A\cap V_\beta. (This is a strengthening, as when letting A = V_\delta, \kappa being A-strong in \delta implies that \kappa is \beta-strong for all \beta, as given any \beta, V_\delta\cap V_\beta=V_\beta must be equal to j(A)\cap V_\beta, V_\delta must be a subset of j(A) and therefore a subset of the range of j.) Finally, a cardinal \delta is Woodin if for any choice of A\subseteq V_\delta, there exists a \kappa such that \kappa is A-strong in \delta. == Consequences ==

Woodin cardinals are important in descriptive set theory. By a result of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is Lebesgue measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset). The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that \Theta _0 is Woodin in the class of hereditarily ordinal-definable sets. \Theta _0 is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)). Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an inner model containing a Woodin cardinal in which there is a \Delta_4^1-well-ordering of the reals, ◊ holds, and the generalized continuum hypothesis holds. Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on \omega_1 is \aleph_2-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an \aleph_1-dense ideal over \aleph_1. ==Hyper-Woodin cardinals==

A cardinal \kappa is called hyper-Woodin if there exists a normal measure U on \kappa such that for every set S, the set :\{\lambda is -S-strong\} is in U. \lambda is -S-strong if and only if for each \delta there is a transitive class N and an elementary embedding :j : V \to N with :\lambda = \text{crit}(j), :j(\lambda) \geq \delta , and :j(S) \cap H_\delta = S \cap H_\delta. The name alludes to the classical result that a cardinal is Woodin if and only if for every set S, the set :\{\lambda is -S-strong\} is a stationary set.p. 363 The measure U will contain the set of all Shelah cardinals below \kappa. ==Weakly hyper-Woodin cardinals==

A cardinal \kappa is called weakly hyper-Woodin if for every set S there exists a normal measure U on \kappa such that the set \{\lambda is -S-strong\} is in U. \lambda is -S-strong if and only if for each \delta there is a transitive class N and an elementary embedding j : V \to N with \lambda = \text{crit}(j), j(\lambda) \geq \delta, and j(S) \cap H_\delta = S \cap H_\delta.p. 3390 The name alludes to the classic result that a cardinal is Woodin if for every set S, the set \{\lambda is -S-strong\} is stationary. The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of U does not depend on the choice of the set S for hyper-Woodin cardinals. ==Woodin-in-the-next-admissible cardinals==

Let \delta be a cardinal and let \alpha be the least admissible ordinal greater than \delta. The cardinal \delta is said to be Woodin-in-the-next-admissible if for any function f:\delta\to\delta such that f\in L_\alpha(V_\delta), there exists \kappa such that f[\kappa]\subseteq\kappa, and there is an extender E\in V_\delta such that \mathrm{crit}(E)=\kappa and V_{i_E(f)(\kappa)}\subset\mathrm{Ult}(V,E). These cardinals appear when building models from iteration trees.p.4 == Notes and references ==