In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number such that for all functions , there exists a cardinal with and an elementary embedding from the Von Neumann universe into a transitive inner model with critical point and .

An equivalent definition is this: is Woodin if and only if is strongly inaccessible and for all there exists a which is --strong.

being --strong means that for all ordinals , there exist a which is an elementary embedding with critical point , , and . (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

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The hierarchy   (known as the von Neumann hierarchy) is defined by transfinite recursion on  :

  •  ,
  •  ,
  •  , when   is a limit ordinal.

For any ordinal  ,   is a set. The union of the sets   for all ordinals   is no longer a set, but a proper class. Some of the sets   have set-theoretic properties, for example when   is an inaccessible cardinal,   satisfies second-order ZFC ("satisfies" here means the notion of satisfaction from first-order logic).

For a transitive class  , a function   is said to be an elementary embedding if for any formula   with free variables   in the language of set theory, it is the case that   iff  , where   is first-order logic's notion of satisfaction as before. An elementary embedding   is called nontrivial if it is not the identity. If   is a nontrivial elementary embedding, there exists an ordinal   such that  , and the least such   is called the critical point of  .

Many large cardinal properties can be phrased in terms of elementary embeddings. For an ordinal  , a cardinal   is said to be  -strong if a transitive class   can be found such that there is a nontrivial elementary embedding   whose critical point is  , and in addition  .

A strengthening of the notion of  -strong cardinal is the notion of  -strongness of a cardinal   in a greater cardinal  : if   and   are cardinals with  , and   is a subset of  , then   is said to be  -strong in   if for all  , there is a nontrivial elementary embedding   witnessing that   is  -strong, and in addition  . (This is a strengthening, as when letting  ,   being  -strong in   implies that   is  -strong for all  , as given any  ,   must be equal to  ,   must be a subset of   and therefore a subset of the range of  .) Finally, a cardinal   is Woodin if for any choice of  , there exists a   such that   is  -strong in  .[1]

Consequences

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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   is Woodin in the class of hereditarily ordinal-definable sets.   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  -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   is  -saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an  -dense ideal over  .

Hyper-Woodin cardinals

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A cardinal   is called hyper-Woodin if there exists a normal measure   on   such that for every set  , the set

  is  - -strong 

is in  .

  is  - -strong if and only if for each   there is a transitive class   and an elementary embedding

 

with

 
 , and
 .

The name alludes to the classical result that a cardinal is Woodin if and only if for every set  , the set

  is  - -strong 

is a stationary set.

The measure   will contain the set of all Shelah cardinals below  .

Weakly hyper-Woodin cardinals

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A cardinal   is called weakly hyper-Woodin if for every set   there exists a normal measure   on   such that the set   is  - -strong  is in  .   is  - -strong if and only if for each   there is a transitive class   and an elementary embedding   with  ,  , and  

The name alludes to the classic result that a cardinal is Woodin if for every set  , the set   is  - -strong  is stationary.

The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of   does not depend on the choice of the set   for hyper-Woodin cardinals.

Woodin-in-the-next-admissible cardinals

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Let   be a cardinal and let   be the least admissible ordinal greater than  . The cardinal   is said to be Woodin-in-the-next-admissible if for any function   such that  , there exists   such that  , and there is an extender   such that   and  . These cardinals appear when building models from iteration trees.[4]p.4

Notes and references

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  1. ^ Steel, John R. (October 2007). "What is a Woodin Cardinal?" (PDF). Notices of the American Mathematical Society. 54 (9): 1146–7. Retrieved 2024-03-04.
  2. ^ 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

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