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woodin cardinal wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 explanation 2 consequences 3 hyper woodin cardinals 4 weakly hyper woodin cardinals 5 woodin in the next admissible cardinals 6 notes and references 7 further reading toggle the table of contents woodin cardinal 2 languages nederlands português edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia kind of large cardinal number this article may be too technical for most readers to understand please help improve it to make it understandable to non experts without removing the technical details february 2023 learn how and when to remove this message in set theory a woodin cardinal named for w hugh woodin is a cardinal number λ displaystyle lambda such that for all functions f λ λ displaystyle f lambda to lambda there exists a cardinal κ λ displaystyle kappa lambda with f β β κ κ displaystyle f beta mid beta kappa subseteq kappa and an elementary embedding j v m displaystyle j v to m from the von neumann universe v displaystyle v into a transitive inner model m displaystyle m with critical point κ displaystyle kappa and v j f κ m displaystyle v_ j f kappa subseteq m an equivalent definition is this λ displaystyle lambda is woodin if and only if λ displaystyle lambda is strongly inaccessible and for all a v λ displaystyle a subseteq v_ lambda there exists a λ a λ displaystyle lambda _ a lambda which is λ displaystyle lambda a displaystyle a strong λ a displaystyle lambda _ a being λ displaystyle lambda a displaystyle a strong means that for all ordinals α λ displaystyle alpha lambda there exist a j v m displaystyle j v to m which is an elementary embedding with critical point λ a displaystyle lambda _ a j λ a α displaystyle j lambda _ a alpha v α m displaystyle v_ alpha subseteq m and j a v α a v α displaystyle 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 1 p 364 explanation edit the hierarchy v α displaystyle v_ alpha known as the von neumann hierarchy is defined by transfinite recursion on α displaystyle alpha v 0 displaystyle v_ 0 varnothing v α 1 p v α displaystyle v_ alpha 1 mathcal p v_ alpha v α β α v β displaystyle v_ alpha bigcup _ beta alpha v_ beta when α displaystyle alpha is a limit ordinal for any ordinal α displaystyle alpha v α displaystyle v_ alpha is a set the union of the sets v α displaystyle v_ alpha for all ordinals α displaystyle alpha is no longer a set but a proper class some of the sets v α displaystyle v_ alpha have set theoretic properties for example when κ displaystyle kappa is an inaccessible cardinal v κ displaystyle v_ kappa satisfies second order zfc satisfies here means the notion of satisfaction from first order logic for a transitive class m displaystyle m a function j v m displaystyle j v to m is said to be an elementary embedding if for any formula ϕ displaystyle phi with free variables x 1 x n displaystyle x_ 1 ldots x_ n in the language of set theory it is the case that v ϕ x 1 x n displaystyle v vdash phi x_ 1 ldots x_ n iff m ϕ j x 1 j x n displaystyle m vdash phi j x_ 1 ldots j x_ n where displaystyle vdash is first order logic s notion of satisfaction as before an elementary embedding j displaystyle j is called nontrivial if it is not the identity if j v m displaystyle j v to m is a nontrivial elementary embedding there exists an ordinal κ displaystyle kappa such that j κ κ displaystyle j kappa neq kappa and the least such κ displaystyle kappa is called the critical point of j displaystyle j many large cardinal properties can be phrased in terms of elementary embeddings for an ordinal β displaystyle beta a cardinal κ displaystyle kappa is said to be β displaystyle beta strong if a transitive class m displaystyle m can be found such that there is a nontrivial elementary embedding j v m displaystyle j v to m whose critical point is κ displaystyle kappa and in addition v β m displaystyle v_ beta subseteq m a strengthening of the notion of β displaystyle beta strong cardinal is the notion of a displaystyle a strongness of a cardinal κ displaystyle kappa in a greater cardinal δ displaystyle delta if κ displaystyle kappa and δ displaystyle delta are cardinals with κ δ displaystyle kappa delta and a displaystyle a is a subset of v δ displaystyle v_ delta then κ displaystyle kappa is said to be a displaystyle a strong in δ displaystyle delta if for all β δ displaystyle beta delta there is a nontrivial elementary embedding j v m displaystyle j v to m witnessing that κ displaystyle kappa is β displaystyle beta strong and in addition j a v β a v β displaystyle j a cap v_ beta a cap v_ beta this is a strengthening as when letting a v δ displaystyle a v_ delta κ displaystyle kappa being a displaystyle a strong in δ displaystyle delta implies that κ displaystyle kappa is β displaystyle beta strong for all β δ displaystyle beta delta as given any β δ displaystyle beta delta v δ v β v β displaystyle v_ delta cap v_ beta v_ beta must be equal to j a v β displaystyle j a cap v_ beta v δ displaystyle v_ delta must be a subset of j a displaystyle j a and therefore a subset of the range of j displaystyle j finally a cardinal δ displaystyle delta is woodin if for any choice of a v δ displaystyle a subseteq v_ delta there exists a κ δ displaystyle kappa delta such that κ displaystyle kappa is a displaystyle a strong in δ displaystyle delta 2 consequences edit woodin cardinals are important in descriptive set theory by a result 3 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 θ 0 displaystyle theta _ 0 is woodin in the class of hereditarily ordinal definable sets θ 0 displaystyle 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 δ 4 1 displaystyle delta _ 4 1 well ordering of the reals holds and the generalized continuum hypothesis holds 4 shelah proved that if the existence of a woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω 1 displaystyle omega _ 1 is ℵ 2 displaystyle aleph _ 2 saturated woodin also proved the equiconsistency of the existence of infinitely many woodin cardinals and the existence of an ℵ 1 displaystyle aleph _ 1 dense ideal over ℵ 1 displaystyle aleph _ 1 hyper woodin cardinals edit a cardinal κ displaystyle kappa is called hyper woodin if there exists a normal measure u displaystyle u on κ displaystyle kappa such that for every set s displaystyle s the set λ κ λ displaystyle lambda kappa mid lambda is κ displaystyle kappa s displaystyle s strong displaystyle is in u displaystyle u λ displaystyle lambda is κ displaystyle kappa s displaystyle s strong if and only if for each δ κ displaystyle delta kappa there is a transitive class n displaystyle n and an elementary embedding j v n displaystyle j v to n with λ crit j displaystyle lambda text crit j j λ δ displaystyle j lambda geq delta and j s h δ s h δ displaystyle 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 displaystyle s the set λ κ λ displaystyle lambda kappa mid lambda is κ displaystyle kappa s displaystyle s strong displaystyle is a stationary set 1 p 363 the measure u displaystyle u will contain the set of all shelah cardinals below κ displaystyle kappa weakly hyper woodin cardinals edit a cardinal κ displaystyle kappa is called weakly hyper woodin if for every set s displaystyle s there exists a normal measure u displaystyle u on κ displaystyle kappa such that the set λ κ λ displaystyle lambda kappa mid lambda is κ displaystyle kappa s displaystyle s strong displaystyle is in u displaystyle u λ displaystyle lambda is κ displaystyle kappa s displaystyle s strong if and only if for each δ κ displaystyle delta kappa there is a transitive class n displaystyle n and an elementary embedding j v n displaystyle j v to n with λ crit j displaystyle lambda text crit j j λ δ displaystyle j lambda geq delta and j s h δ s h δ displaystyle j s cap h_ delta s cap h_ delta 5 p 3390 the name alludes to the classic result that a cardinal is woodin if for every set s displaystyle s the set λ κ λ displaystyle lambda kappa mid lambda is κ displaystyle kappa s displaystyle s strong displaystyle is stationary the difference between hyper woodin cardinals and weakly hyper woodin cardinals is that the choice of u displaystyle u does not depend on the choice of the set s displaystyle s for hyper woodin cardinals woodin in the next admissible cardinals edit let δ displaystyle delta be a cardinal and let α displaystyle alpha be the least admissible ordinal greater than δ displaystyle delta the cardinal δ displaystyle delta is said to be woodin in the next admissible if for any function f δ δ displaystyle f delta to delta such that f l α v δ displaystyle f in l_ alpha v_ delta there exists κ δ displaystyle kappa delta such that f κ κ displaystyle f kappa subseteq kappa and there is an extender e v δ displaystyle e in v_ delta such that c r i t e κ displaystyle mathrm crit e kappa and v i e f κ u l t v e displaystyle v_ i_ e f kappa subset mathrm ult v e these cardinals appear when building models from iteration trees 6 p 4 notes and references edit 1 2 a kanamori the higher infinite large cardinals in set theory from their beginnings isbn 978 3 540 88866 6 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 d a martin j r steel a proof of projective determinacy journal of the american mathematical society vol 2 no 1 1989 w mitchell inner models for large cardinals 2012 p 32 accessed 2022 12 08 e schimmerling woodin caridnals shelah cardinals and the mitchell steel core model proceeding of the american mathematical society vol 130 no 11 2002 a andretta large cardinals and iteration trees of height ω annals of pure and applied logic vol 54 1990 pp 1 15 further reading edit 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 retrieved from https en wikipedia org w index php title woodin_cardinal oldid 1370618123 categories large cardinals determinacy hidden categories articles with short description short description matches wikidata wikipedia articles that are too technical from february 2023 all articles that are too technical this page was last edited on 22 august 2026 at 03 13 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents woodin cardinal 2 languages add topic