Woodin cardinal

In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number [math]\displaystyle{ \lambda }[/math] such that for all functions

[math]\displaystyle{ f : \lambda \to \lambda }[/math]

there exists a cardinal [math]\displaystyle{ \kappa \lt \lambda }[/math] with

[math]\displaystyle{ \{f(\beta) \mid \beta \lt \kappa \} \subseteq \kappa }[/math]

and an elementary embedding

[math]\displaystyle{ j : V \to M }[/math]

from the Von Neumann universe [math]\displaystyle{ V }[/math] into a transitive inner model [math]\displaystyle{ M }[/math] with critical point [math]\displaystyle{ \kappa }[/math] and

[math]\displaystyle{ V_{j(f)(\kappa)} \subseteq M. }[/math]

An equivalent definition is this: [math]\displaystyle{ \lambda }[/math] is Woodin if and only if [math]\displaystyle{ \lambda }[/math] is strongly inaccessible and for all [math]\displaystyle{ A \subseteq V_\lambda }[/math] there exists a [math]\displaystyle{ \lambda_A \lt \lambda }[/math] which is [math]\displaystyle{ \lt \lambda }[/math]-[math]\displaystyle{ A }[/math]-strong.

[math]\displaystyle{ \lambda _A }[/math] being [math]\displaystyle{ \lt \lambda }[/math]-[math]\displaystyle{ A }[/math]-strong means that for all ordinals [math]\displaystyle{ \alpha \lt \lambda }[/math], there exist a [math]\displaystyle{ j: V \to M }[/math] which is an elementary embedding with critical point [math]\displaystyle{ \lambda _A }[/math], [math]\displaystyle{ j(\lambda _A) \gt \alpha }[/math], [math]\displaystyle{ V_\alpha \subseteq M }[/math] and [math]\displaystyle{ 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]\displaystyle{ \Theta _0 }[/math] is Woodin in the class of hereditarily ordinal-definable sets. [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \omega_1 }[/math] is [math]\displaystyle{ \aleph_2 }[/math]-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an [math]\displaystyle{ \aleph_1 }[/math]-dense ideal over [math]\displaystyle{ \aleph_1 }[/math].

Hyper-Woodin cardinals

A cardinal [math]\displaystyle{ \kappa }[/math] is called hyper-Woodin if there exists a normal measure [math]\displaystyle{ U }[/math] on [math]\displaystyle{ \kappa }[/math] such that for every set [math]\displaystyle{ S }[/math], the set

[math]\displaystyle{ \{\lambda \lt \kappa \mid \lambda }[/math] is [math]\displaystyle{ \lt \kappa }[/math]-[math]\displaystyle{ S }[/math]-strong[math]\displaystyle{ \} }[/math]

is in [math]\displaystyle{ U }[/math].

[math]\displaystyle{ \lambda }[/math] is [math]\displaystyle{ \lt \kappa }[/math]-[math]\displaystyle{ S }[/math]-strong if and only if for each [math]\displaystyle{ \delta \lt \kappa }[/math] there is a transitive class [math]\displaystyle{ N }[/math] and an elementary embedding

[math]\displaystyle{ j : V \to N }[/math]

with

[math]\displaystyle{ \lambda = \text{crit}(j), }[/math]

[math]\displaystyle{ j(\lambda) \geq \delta }[/math], and

[math]\displaystyle{ 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]\displaystyle{ S }[/math], the set

[math]\displaystyle{ \{\lambda \lt \kappa \mid \lambda }[/math] is [math]\displaystyle{ \lt \kappa }[/math]-[math]\displaystyle{ S }[/math]-strong[math]\displaystyle{ \} }[/math]

is a stationary set.

The measure [math]\displaystyle{ U }[/math] will contain the set of all Shelah cardinals below [math]\displaystyle{ \kappa }[/math].

Weakly hyper-Woodin cardinals

A cardinal [math]\displaystyle{ \kappa }[/math] is called weakly hyper-Woodin if for every set [math]\displaystyle{ S }[/math] there exists a normal measure [math]\displaystyle{ U }[/math] on [math]\displaystyle{ \kappa }[/math] such that the set [math]\displaystyle{ \{\lambda \lt \kappa \mid \lambda }[/math] is [math]\displaystyle{ \lt \kappa }[/math]-[math]\displaystyle{ S }[/math]-strong[math]\displaystyle{ \} }[/math] is in [math]\displaystyle{ U }[/math]. [math]\displaystyle{ \lambda }[/math] is [math]\displaystyle{ \lt \kappa }[/math]-[math]\displaystyle{ S }[/math]-strong if and only if for each [math]\displaystyle{ \delta \lt \kappa }[/math] there is a transitive class [math]\displaystyle{ N }[/math] and an elementary embedding [math]\displaystyle{ j : V \to N }[/math] with [math]\displaystyle{ \lambda = \text{crit}(j) }[/math], [math]\displaystyle{ j(\lambda) \geq \delta }[/math], and [math]\displaystyle{ j(S) \cap H_\delta = S \cap H_\delta. }[/math]

The name alludes to the classic result that a cardinal is Woodin if for every set [math]\displaystyle{ S }[/math], the set [math]\displaystyle{ \{\lambda \lt \kappa \mid \lambda }[/math] is [math]\displaystyle{ \lt \kappa }[/math]-[math]\displaystyle{ S }[/math]-strong[math]\displaystyle{ \} }[/math] is stationary.

The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of [math]\displaystyle{ U }[/math] does not depend on the choice of the set [math]\displaystyle{ S }[/math] for hyper-Woodin cardinals.

Further reading

• Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3. 
• For proofs of the two results listed in consequences see Handbook of Set Theory (Eds. Foreman, Kanamori, Magidor) (to appear). Drafts of some chapters are available.
• Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002, online
• Steel, John R. (October 2007). "What is a Woodin Cardinal?". Notices of the American Mathematical Society 54 (9): 1146–7. http://www.ams.org/notices/200709/tx070901146p.pdf. Retrieved 2008-01-15.