Woodin cardinal
Woodin cardinal
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number <math>\lambda</math> such that for all functions <math>f : \lambda \to \lambda</math>, there exists a cardinal <math>\kappa < \lambda</math> with <math>\{f(\beta) \mid \beta < \kappa \} \subseteq \kappa</math> and an elementary embedding <math>j : V \to M</math> from the Von Neumann universe <math>V</math> into a transitive inner model <math>M</math> with critical point <math>\kappa</math> and <math>V_{j(f)(\kappa)} \subseteq M</math>.
An equivalent definition is this: <math>\lambda</math> is Woodin if and only if <math>\lambda</math> is strongly inaccessible and for all <math>A \subseteq V_\lambda</math> there exists a <math>\lambda_A < \lambda</math> which is <math><\lambda</math>-<math>A</math>-strong.
<math>\lambda _A</math> being <math><\lambda</math>-<math>A</math>-strong means that for all ordinals <math>\alpha < \lambda </math>, there exist a <math>j: V \to M</math> which is an elementary embedding with critical point <math>\lambda _A</math>, <math>j(\lambda _A) > \alpha</math>, <math>V_\alpha \subseteq M</math> and <math>j(A) \cap V_\alpha = A \cap V_\alpha</math>. (See also strong cardinal.)
A Woodin cardinal is preceded by a stationary set of measurable-cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact.
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 <math>\Theta _0</math> is Woodin in the class of hereditarily ordinal-definable sets. <math>\Theta _0</math> 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 <math>\Delta_4^1</math>-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 <math>\omega_1</math> is <math>\aleph_2</math>-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an <math>\aleph_1</math>-dense ideal over <math>\aleph_1</math>.
Hyper-Woodin cardinals A cardinal <math>\kappa</math> is called hyper-Woodin if there exists a normal measure <math>U</math> on <math>\kappa</math> such that for every set <math>S</math>, the set
:<math>\{\lambda < \kappa \mid \lambda</math> is <math>< \kappa</math>-<math>S</math>-strong<math>\}</math>
is in <math>U</math>.
<math>\lambda</math> is <math><\kappa</math>-<math>S</math>-strong if and only if for each <math>\delta < \kappa</math> there is a transitive class <math>N</math> and an elementary embedding
:<math>j : V \to N</math>
with
:<math>\lambda = \text{crit}(j),</math> :<math>j(\lambda) \geq \delta </math>, and
:<math>j(S) \cap H_\delta = S \cap H_\delta</math>.
The name alludes to the classical result that a cardinal is Woodin if and only if for every set <math>S</math>, the set
:<math>\{\lambda < \kappa \mid \lambda</math> is <math>< \kappa</math>-<math>S</math>-strong<math>\}</math>
is a stationary set.p. 363
The measure <math>U</math> will contain the set of all Shelah cardinals below <math>\kappa</math>.
Weakly hyper-Woodin cardinals A cardinal <math>\kappa</math> is called weakly hyper-Woodin if for every set <math>S</math> there exists a normal measure <math>U</math> on <math>\kappa</math> such that the set <math>\{\lambda < \kappa \mid \lambda</math> is <math>< \kappa</math>-<math>S</math>-strong<math>\}</math> is in <math>U</math>. <math>\lambda</math> is <math><\kappa</math>-<math>S</math>-strong if and only if for each <math>\delta < \kappa</math> there is a transitive class <math>N</math> and an elementary embedding <math>j : V \to N</math> with <math>\lambda = \text{crit}(j)</math>, <math>j(\lambda) \geq \delta</math>, and <math>j(S) \cap H_\delta = S \cap H_\delta.</math>p. 3390
The name alludes to the classic result that a cardinal is Woodin if for every set <math>S</math>, the set <math>\{\lambda < \kappa \mid \lambda</math> is <math>< \kappa</math>-<math>S</math>-strong<math>\}</math> is stationary.
The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of <math>U</math> does not depend on the choice of the set <math>S</math> for hyper-Woodin cardinals.