Woodin cardinal explained

λ such that for all functions f : λ \to λ , there exists a cardinal \kappa < λ with \{f(\beta) \mid \beta < \kappa \} \subseteq \kappa and an elementary embedding j : V \to M from the Von Neumann universe V into a transitive inner model M with critical point \kappa and V _{j(f)(\kappa)} \subseteq M .

An equivalent definition is this:

λ is Woodin if and only if λ is strongly inaccessible and for all A \subseteq V _{λ} there exists a λ _{A} < λ which is < λ -A -strong.λ _{A} being < λ -A -strong means that for all ordinals \alpha < λ , there exist a j: V \to M which is an elementary embedding with critical point λ _{A} , j( λ _{A)} > \alpha , V _{\alpha} \subseteq M and j(A) \cap V _{\alpha} = A \cap V _{\alpha} . (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.

Explanation

The hierarchy

V _{\alpha} (known as the von Neumann hierarchy) is defined by transfinite recursion on \alpha :V _{0} = \varnothing ,V _{\alpha+1} = l P(V _{\alpha)} ,V _{\alpha} = c up _{\beta<\alpha} V _{\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).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) ≠ \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<\delta , and A is a subset of V _{\delta} , then \kappa is said to be A -strong in \delta if for all \beta<\delta , 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<\delta , as given any \beta<\delta , 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<\delta such that \kappa is A -strong in \delta .^[1]

Consequences

Woodin cardinals are important in descriptive set theory. By a result^[2] 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 ^{1} _{4} -well-ordering of the reals, ◊ holds, and the generalized continuum hypothesis holds.^[3]

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

\kappa is called hyper-Woodin if there exists a normal measure U on \kappa such that for every set S , the set\{ λ < \kappa \mid λ is < \kappa -S -strong\}

is in

U .λ is <\kappa -S -strong if and only if for each \delta < \kappa there is a transitive class N and an elementary embeddingj : V \to N

with

λ = crit (j), j( λ ) \geq \delta , andj(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\{ λ < \kappa \mid λ is < \kappa -S -strong\}

is a stationary set.

The measure

U will contain the set of all Shelah cardinals below \kappa .

Weakly hyper-Woodin cardinals

\kappa is called weakly hyper-Woodin if for every set S there exists a normal measure U on \kappa such that the set \{ λ < \kappa \mid λ is < \kappa -S -strong\} is in U . λ is <\kappa -S -strong if and only if for each \delta < \kappa there is a transitive class N and an elementaryembedding j : V \to N with λ = crit (j) , j( λ ) \geq \delta , and j(S) \cap H _{\delta} = S \cap H _{\delta.}

The name alludes to the classic result that a cardinal is Woodin if for every set

S , the set \{ λ < \kappa \mid λ is < \kappa -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<\delta such that f[\kappa]\subseteq\kappa , and there is an extender E\in V _{\delta} such that crit (E) = \kappa and V _{i _{E(f)(\kappa)}} \subset Ult (V,E) . These cardinals appear when building models from iteration trees.^{[4] p.4}

Notes and references

1. Steel . John R. . John R. Steel . October 2007 . What is a Woodin Cardinal? . . 54 . 9 . 1146 - 7 . 2024-03-04 .
2. https://www.jstor.org/stable/1990913 A Proof of Projective Determinacy
3. W. Mitchell, Inner models for large cardinals (2012, p.32). Accessed 2022-12-08.
4. A. Andretta, "Large cardinals and iteration trees of height ω", Annals of Pure and Applied Logic vol. 54 (1990), pp.1--15.

Further reading

• Book: Kanamori, Akihiro. Akihiro Kanamori

. 2003. Akihiro Kanamori. Springer. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite. 2nd. 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