{"history":[],"article":{"id":"b45e68a4-79e3-4a83-8c03-41b358cb9d22","title":"Inaccessible cardinal","slug":"inaccessible-cardinal","content":":::summary\n:::\n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), an [uncountable](https://bliptext.com/articles/uncountable-set) [cardinal](https://bliptext.com/articles/cardinal-number) is **inaccessible** if it cannot be obtained from smaller [cardinal](https://bliptext.com/articles/cardinal-number)s by the usual operations of [cardinal](https://bliptext.com/articles/cardinal-number) arithmetic.\nMore precisely, a cardinal κ is **strongly inaccessible** if it satisfies the following three conditions: it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and implies.\nThe term \"inaccessible cardinal\" is ambiguous.\nUntil about 1950, it meant \"weakly inaccessible cardinal\", but since then it usually means \"strongly inaccessible cardinal\".\nAn uncountable cardinal is **weakly inaccessible** if it is a [regular](https://bliptext.com/articles/regular-cardinal) [weak limit cardinal](https://bliptext.com/articles/weak-limit-cardinal).\nIt is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above).\nSome authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case \\aleph_0 is strongly inaccessible).\nWeakly inaccessible cardinals were introduced by, and strongly inaccessible ones by and ; in the latter they were referred to along with \\aleph_0 as Grenzzahlen ([English](https://bliptext.com/articles/english-language) \"limit numbers\").\nEvery strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal.\nIf the [generalized continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.\n\\aleph_0 ([aleph-null](https://bliptext.com/articles/aleph-number)) is a regular strong limit cardinal.\nAssuming the [axiom of choice](https://bliptext.com/articles/axiom-of-choice), every other infinite cardinal number is regular or a (weak) limit.\nHowever, only a rather large cardinal number can be both and thus weakly inaccessible.\nAn [ordinal](https://bliptext.com/articles/ordinal-number) is a weakly inaccessible cardinal if and only if it is a regular [ordinal](https://bliptext.com/articles/ordinal-number) and it is a limit of regular [ordinal](https://bliptext.com/articles/ordinal-number)s.\n(Zero, one, and ω are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible.\nThe assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a [Grothendieck universe](https://bliptext.com/articles/grothendieck-universe), the two ideas being intimately connected.\n\n# Models and consistency\nSuppose that \\kappa is a cardinal number.\n[Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory) with Choice (ZFC) implies that the \\kappath level of the [Von Neumann universe](https://bliptext.com/articles/von-neumann-universe) V_\\kappa is a [model](https://bliptext.com/articles/model-theory) of ZFC whenever \\kappa is strongly inaccessible.\nFurthermore, ZF implies that the [Gödel universe](https://bliptext.com/articles/g-del-s-constructible-universe) L_\\kappa is a model of ZFC whenever \\kappa is weakly inaccessible.\nThus, ZF together with \"there exists a weakly inaccessible cardinal\" implies that ZFC is consistent.\nTherefore, inaccessible cardinals are a type of [large cardinal](https://bliptext.com/articles/large-cardinal).\nIf V is a standard model of ZFC and \\kappa is an inaccessible in V, then\nHere, Def(X) is the set of Δ0-definable subsets of X (see [constructible universe](https://bliptext.com/articles/constructible-universe)).\nIt is worth pointing out that the first claim can be weakened: \\kappa does not need to be inaccessible, or even a cardinal number, in order for\nV\\kappa\nto be a standard model of ZF (see [below](https://bliptext.com/articles/inaccessible-cardinal)).\nSuppose V is a model of ZFC.\nEither V contains no strong inaccessible or, taking \\kappa to be the smallest strong inaccessible in V, V_\\kappa is a standard model of ZFC which contains no strong inaccessibles.\nThus, the consistency of ZFC implies consistency of ZFC+\"there are no strong inaccessibles\".\nSimilarly, either V contains no weak inaccessible or, taking \\kappa to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then L_\\kappa is a standard model of ZFC which contains no weak inaccessibles.\nSo consistency of ZFC implies consistency of ZFC+\"there are no weak inaccessibles\".\nThis shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals.\nThe issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle.\nThe proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + \"there is not an inaccessible cardinal\" can be formalized in ZFC.\nHowever, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + \"there is an inaccessible cardinal\" can be formalized in ZFC.\nThis follows from [Gödel's second incompleteness theorem](https://bliptext.com/articles/g-del-s-second-incompleteness-theorem), which shows that if ZFC + \"there is an inaccessible cardinal\" is consistent, then it cannot prove its own consistency.\nBecause ZFC + \"there is an inaccessible cardinal\" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + \"there is an inaccessible cardinal\" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent.\nThere are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC.\nOne such argument, presented by, is that the class of all ordinals of a particular model M of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending M and preserving powerset of elements of M.\n\n# Existence of a proper class of inaccessibles\nThere are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest.\nIn the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger,\nμ < κ\n. Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom).\nAs is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC.\nAssuming ZFC, the inaccessible cardinal axiom is equivalent to the **universe axiom** of [Grothendieck](https://bliptext.com/articles/grothendieck) and [Verdier](https://bliptext.com/articles/jean-louis-verdier): every set is contained in a [Grothendieck](https://bliptext.com/articles/grothendieck) universe.\nThe axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with [urelements](https://bliptext.com/articles/urelement)).\nThis axiomatic system is useful to prove for example that every [category](https://bliptext.com/articles/category-mathematics) has an appropriate [Yoneda embedding](https://bliptext.com/articles/yoneda-embedding).\nThis is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e. the class of all ordinals in your model.\n\n# α-inaccessible cardinals and hyper-inaccessible cardinals\nThe term \"α-inaccessible cardinal\" is ambiguous and different authors use inequivalent definitions.\nOne definition is that\na cardinal κ is called α-inaccessible, for any ordinal α, if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular).\nIn this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals.\nAnother possible definition is that a cardinal κ is called α-weakly inaccessible if κ is regular and for every ordinal β < α, the set of β-weakly inaccessibles less than κ is unbounded in κ.\nIn this case the 0-weakly inaccessible cardinals are the regular cardinals and the 1-weakly inaccessible cardinals are the weakly inaccessible cardinals.\nThe α-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles.\nFor example, denote by ψ0(λ) the λth inaccessible cardinal, then the fixed points of ψ0 are the 1-inaccessible cardinals.\nThen letting ψβ(λ) be the λth β-inaccessible cardinal, the fixed points of ψβ are the (β+1)-inaccessible cardinals (the values ψβ+1(λ)).\nIf α is a limit ordinal, an α-inaccessible is a fixed point of every ψβ for β < α (the value ψα(λ) is the λth such cardinal).\nThis process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of [large cardinal numbers](https://bliptext.com/articles/list-of-large-cardinal-properties).\nThe term **hyper-inaccessible** is ambiguous and has at least three incompatible meanings.\nMany authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible).\nOther authors use it to mean that κ is κ-inaccessible.\n(It can never be\nκ+1\n-inaccessible.) It is occasionally used to mean [Mahlo cardinal](https://bliptext.com/articles/mahlo-cardinal).\nThe term α-hyper-inaccessible is also ambiguous.\nSome authors use it to mean α-inaccessible.\nOther authors use the definition that\nfor any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ.\nHyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous.\nUsing \"weakly inaccessible\" instead of \"inaccessible\", similar definitions can be made for \"weakly α-inaccessible\", \"weakly hyper-inaccessible\", and \"weakly α-hyper-inaccessible\".\n[Mahlo cardinals](https://bliptext.com/articles/mahlo-cardinal) are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.\n\n# Two model-theoretic characterisations of inaccessibility\nFirstly, a cardinal κ is inaccessible if and only if κ has the following [reflection](https://bliptext.com/articles/reflection-principle) property: for all subsets, there exists such that is an [elementary substructure](https://bliptext.com/articles/elementary-substructure) of.\n(In fact, the set of such α is [closed unbounded](https://bliptext.com/articles/club-set) in κ.) Therefore, \\kappa is \\Pi_n^0-[indescribable](https://bliptext.com/articles/totally-indescribable-cardinal) for all n ≥ 0.\nOn the other hand, there is not necessarily an ordinal such that V_\\kappa, and if this holds, then \\kappa must be the \\kappath inaccessible cardinal.\nIt is provable in ZF that V has a somewhat weaker reflection property, where the substructure is only required to be 'elementary' with respect to a finite set of formulas.\nUltimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation\n⊧\ncan be defined, semantic truth itself (i.e. \\vDash_V) cannot, due to [Tarski's theorem](https://bliptext.com/articles/tarski-s-undefinability-theorem).\nSecondly, under ZFC [Zermelo's categoricity theorem](https://bliptext.com/articles/zermelo-s-categoricity-theorem) can be shown, which states that \\kappa is inaccessible if and only if is a model of [second order](https://bliptext.com/articles/second-order-logic) ZFC.\nIn this case, by the reflection property above, there exists such that is a standard model of ([first order](https://bliptext.com/articles/first-order-logic)) ZFC.\nHence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a transitive model of ZFC.\nInaccessibility of \\kappa is a \\Pi^1_1 property over V_\\kappa, while a cardinal \\pi being inaccessible (in some given model of \\mathrm{ZF} containing \\pi) is \\Pi_1.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:52:25.713Z","updated_at":"2025-01-25T18:52:25.713Z","search_vector":"'cardin':2 'inaccess':1","search_content":" \n \n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), an [uncountable](https://bliptext.com/articles/uncountable-set) [cardinal](https://bliptext.com/articles/cardinal-number) is **inaccessible** if it cannot be obtained from smaller [cardinal](https://bliptext.com/articles/cardinal-number)s by the usual operations of [cardinal](https://bliptext.com/articles/cardinal-number) arithmetic.\nMore precisely, a cardinal κ is **strongly inaccessible** if it satisfies the following three conditions: it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and implies.\nThe term \"inaccessible cardinal\" is ambiguous.\nUntil about 1950, it meant \"weakly inaccessible cardinal\", but since then it usually means \"strongly inaccessible cardinal\".\nAn uncountable cardinal is **weakly inaccessible** if it is a [regular](https://bliptext.com/articles/regular-cardinal) [weak limit cardinal](https://bliptext.com/articles/weak-limit-cardinal).\nIt is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above).\nSome authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case \\aleph_0 is strongly inaccessible).\nWeakly inaccessible cardinals were introduced by, and strongly inaccessible ones by and ; in the latter they were referred to along with \\aleph_0 as Grenzzahlen ([English](https://bliptext.com/articles/english-language) \"limit numbers\").\nEvery strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal.\nIf the [generalized continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.\n\\aleph_0 ([aleph-null](https://bliptext.com/articles/aleph-number)) is a regular strong limit cardinal.\nAssuming the [axiom of choice](https://bliptext.com/articles/axiom-of-choice), every other infinite cardinal number is regular or a (weak) limit.\nHowever, only a rather large cardinal number can be both and thus weakly inaccessible.\nAn [ordinal](https://bliptext.com/articles/ordinal-number) is a weakly inaccessible cardinal if and only if it is a regular [ordinal](https://bliptext.com/articles/ordinal-number) and it is a limit of regular [ordinal](https://bliptext.com/articles/ordinal-number)s.\n(Zero, one, and ω are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible.\nThe assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a [Grothendieck universe](https://bliptext.com/articles/grothendieck-universe), the two ideas being intimately connected.\n\n# Models and consistency\nSuppose that \\kappa is a cardinal number.\n[Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory) with Choice (ZFC) implies that the \\kappath level of the [Von Neumann universe](https://bliptext.com/articles/von-neumann-universe) V_\\kappa is a [model](https://bliptext.com/articles/model-theory) of ZFC whenever \\kappa is strongly inaccessible.\nFurthermore, ZF implies that the [Gödel universe](https://bliptext.com/articles/g-del-s-constructible-universe) L_\\kappa is a model of ZFC whenever \\kappa is weakly inaccessible.\nThus, ZF together with \"there exists a weakly inaccessible cardinal\" implies that ZFC is consistent.\nTherefore, inaccessible cardinals are a type of [large cardinal](https://bliptext.com/articles/large-cardinal).\nIf V is a standard model of ZFC and \\kappa is an inaccessible in V, then\nHere, Def(X) is the set of Δ0-definable subsets of X (see [constructible universe](https://bliptext.com/articles/constructible-universe)).\nIt is worth pointing out that the first claim can be weakened: \\kappa does not need to be inaccessible, or even a cardinal number, in order for\nV\\kappa\nto be a standard model of ZF (see [below](https://bliptext.com/articles/inaccessible-cardinal)).\nSuppose V is a model of ZFC.\nEither V contains no strong inaccessible or, taking \\kappa to be the smallest strong inaccessible in V, V_\\kappa is a standard model of ZFC which contains no strong inaccessibles.\nThus, the consistency of ZFC implies consistency of ZFC+\"there are no strong inaccessibles\".\nSimilarly, either V contains no weak inaccessible or, taking \\kappa to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then L_\\kappa is a standard model of ZFC which contains no weak inaccessibles.\nSo consistency of ZFC implies consistency of ZFC+\"there are no weak inaccessibles\".\nThis shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals.\nThe issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle.\nThe proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + \"there is not an inaccessible cardinal\" can be formalized in ZFC.\nHowever, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + \"there is an inaccessible cardinal\" can be formalized in ZFC.\nThis follows from [Gödel's second incompleteness theorem](https://bliptext.com/articles/g-del-s-second-incompleteness-theorem), which shows that if ZFC + \"there is an inaccessible cardinal\" is consistent, then it cannot prove its own consistency.\nBecause ZFC + \"there is an inaccessible cardinal\" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + \"there is an inaccessible cardinal\" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent.\nThere are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC.\nOne such argument, presented by, is that the class of all ordinals of a particular model M of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending M and preserving powerset of elements of M.\n\n# Existence of a proper class of inaccessibles\nThere are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest.\nIn the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger,\nμ < κ\n. Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom).\nAs is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC.\nAssuming ZFC, the inaccessible cardinal axiom is equivalent to the **universe axiom** of [Grothendieck](https://bliptext.com/articles/grothendieck) and [Verdier](https://bliptext.com/articles/jean-louis-verdier): every set is contained in a [Grothendieck](https://bliptext.com/articles/grothendieck) universe.\nThe axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with [urelements](https://bliptext.com/articles/urelement)).\nThis axiomatic system is useful to prove for example that every [category](https://bliptext.com/articles/category-mathematics) has an appropriate [Yoneda embedding](https://bliptext.com/articles/yoneda-embedding).\nThis is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e. the class of all ordinals in your model.\n\n# α-inaccessible cardinals and hyper-inaccessible cardinals\nThe term \"α-inaccessible cardinal\" is ambiguous and different authors use inequivalent definitions.\nOne definition is that\na cardinal κ is called α-inaccessible, for any ordinal α, if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular).\nIn this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals.\nAnother possible definition is that a cardinal κ is called α-weakly inaccessible if κ is regular and for every ordinal β < α, the set of β-weakly inaccessibles less than κ is unbounded in κ.\nIn this case the 0-weakly inaccessible cardinals are the regular cardinals and the 1-weakly inaccessible cardinals are the weakly inaccessible cardinals.\nThe α-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles.\nFor example, denote by ψ0(λ) the λth inaccessible cardinal, then the fixed points of ψ0 are the 1-inaccessible cardinals.\nThen letting ψβ(λ) be the λth β-inaccessible cardinal, the fixed points of ψβ are the (β+1)-inaccessible cardinals (the values ψβ+1(λ)).\nIf α is a limit ordinal, an α-inaccessible is a fixed point of every ψβ for β < α (the value ψα(λ) is the λth such cardinal).\nThis process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of [large cardinal numbers](https://bliptext.com/articles/list-of-large-cardinal-properties).\nThe term **hyper-inaccessible** is ambiguous and has at least three incompatible meanings.\nMany authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible).\nOther authors use it to mean that κ is κ-inaccessible.\n(It can never be\nκ+1\n-inaccessible.) It is occasionally used to mean [Mahlo cardinal](https://bliptext.com/articles/mahlo-cardinal).\nThe term α-hyper-inaccessible is also ambiguous.\nSome authors use it to mean α-inaccessible.\nOther authors use the definition that\nfor any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ.\nHyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous.\nUsing \"weakly inaccessible\" instead of \"inaccessible\", similar definitions can be made for \"weakly α-inaccessible\", \"weakly hyper-inaccessible\", and \"weakly α-hyper-inaccessible\".\n[Mahlo cardinals](https://bliptext.com/articles/mahlo-cardinal) are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.\n\n# Two model-theoretic characterisations of inaccessibility\nFirstly, a cardinal κ is inaccessible if and only if κ has the following [reflection](https://bliptext.com/articles/reflection-principle) property: for all subsets, there exists such that is an [elementary substructure](https://bliptext.com/articles/elementary-substructure) of.\n(In fact, the set of such α is [closed unbounded](https://bliptext.com/articles/club-set) in κ.) Therefore, \\kappa is \\Pi_n^0-[indescribable](https://bliptext.com/articles/totally-indescribable-cardinal) for all n ≥ 0.\nOn the other hand, there is not necessarily an ordinal such that V_\\kappa, and if this holds, then \\kappa must be the \\kappath inaccessible cardinal.\nIt is provable in ZF that V has a somewhat weaker reflection property, where the substructure is only required to be 'elementary' with respect to a finite set of formulas.\nUltimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation\n⊧\ncan be defined, semantic truth itself (i.e. \\vDash_V) cannot, due to [Tarski's theorem](https://bliptext.com/articles/tarski-s-undefinability-theorem).\nSecondly, under ZFC [Zermelo's categoricity theorem](https://bliptext.com/articles/zermelo-s-categoricity-theorem) can be shown, which states that \\kappa is inaccessible if and only if is a model of [second order](https://bliptext.com/articles/second-order-logic) ZFC.\nIn this case, by the reflection property above, there exists such that is a standard model of ([first order](https://bliptext.com/articles/first-order-logic)) ZFC.\nHence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a transitive model of ZFC.\nInaccessibility of \\kappa is a \\Pi^1_1 property over V_\\kappa, while a cardinal \\pi being inaccessible (in some given model of \\mathrm{ZF} containing \\pi) is \\Pi_1.","revisionCount":0},"hasMore":false,"nextCursor":null}

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