{"history":[],"article":{"id":"be6dc1f8-6d3a-4737-8cad-31dfe6842e3f","title":"Measurable cardinal","slug":"measurable-cardinal","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **measurable cardinal** is a certain kind of [large cardinal](https://bliptext.com/articles/large-cardinal) number.\nIn order to define the concept, one introduces a two-valued [measure](https://bliptext.com/articles/measure-mathematics) on a cardinal κ, or more generally on any set.\nFor a cardinal κ, it can be described as a subdivision of all of its [subsets](https://bliptext.com/articles/subset) into large and small sets such that κ itself is large, ∅ and all [singletons](https://bliptext.com/articles/singleton-mathematics) {α} (with α ∈ κ) are small, [complements](https://bliptext.com/articles/set-complement) of small sets are large and vice versa.\nThe [intersection](https://bliptext.com/articles/intersection) of fewer than κ large sets is again large.\nIt turns out that [uncountable](https://bliptext.com/articles/uncountable) cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from [ZFC](https://bliptext.com/articles/zfc).\nThe concept of a measurable cardinal was introduced by [Stanisław Ulam](https://bliptext.com/articles/stanis-aw-ulam) in 1930.\n\n# Definition\nFormally, a measurable cardinal is an uncountable [cardinal number](https://bliptext.com/articles/cardinal-number) κ such that there exists a κ-additive, non-trivial, 0-1-valued [measure](https://bliptext.com/articles/measure-mathematics) μ on the [power set](https://bliptext.com/articles/power-set) of κ.\nHere, κ-additive means: For every λ < κ and every λ-sized set {Aβ}β<λ of pairwise disjoint subsets Aβ ⊆ κ, we have\nEquivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal [ultrafilter](https://bliptext.com/articles/ultrafilter).\nThis means that the intersection of any strictly less than κ-many sets in the ultrafilter, is also in the ultrafilter.\nEquivalently, κ is measurable means that it is the [critical point](https://bliptext.com/articles/critical-point-set-theory) of a non-trivial [elementary embedding](https://bliptext.com/articles/elementary-embedding) of the [universe](https://bliptext.com/articles/universe-set-theory) V into a [transitive class](https://bliptext.com/articles/transitive-class) M.\nThis equivalence is due to [Jerome Keisler](https://bliptext.com/articles/jerome-keisler) and [Dana Scott](https://bliptext.com/articles/dana-scott), and uses the [ultrapower](https://bliptext.com/articles/ultraproduct) construction from [model theory](https://bliptext.com/articles/model-theory).\nSince V is a [proper class](https://bliptext.com/articles/proper-class), a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called [Scott's trick](https://bliptext.com/articles/scott-s-trick).\n\n# Properties\nIt is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be [regular](https://bliptext.com/articles/regular-cardinal).\n(By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2λ.\nIf this were the case, we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1.\nThe intersection of these λ-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure.\nThus, assuming the Axiom of Choice, we can infer that κ is a [strong limit cardinal](https://bliptext.com/articles/strong-limit-cardinal), which completes the proof of its [inaccessibility](https://bliptext.com/articles/inaccessible-cardinal).\nAlthough it follows from [ZFC](https://bliptext.com/articles/zfc) that every measurable cardinal is [inaccessible](https://bliptext.com/articles/inaccessible-cardinal) (and is [ineffable](https://bliptext.com/articles/ineffable-cardinal), [Ramsey](https://bliptext.com/articles/ramsey-cardinal), etc.), it is consistent with [ZF](https://bliptext.com/articles/zermelo-fraenkel-set-theory) that a measurable cardinal can be a [successor cardinal](https://bliptext.com/articles/successor-cardinal).\nIt follows from ZF + [AD](https://bliptext.com/articles/axiom-of-determinacy) that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a [closed and unbounded](https://bliptext.com/articles/club-set) subset.\nUlam showed that the smallest cardinal κ that admits a non-trivial countably-additive two-valued measure must in fact admit a κ-additive measure.\n(If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.\nIf κ is measurable and p ∈ Vκ and M (the ultrapower of V) satisfies ψ(κ, p), then the set of α < κ such that V satisfies ψ(α, p) is stationary in κ (actually a set of measure 1).\nIn particular if ψ is a Π1 formula and V satisfies ψ(κ, p), then M satisfies it and thus V satisfies ψ(α, p) for a stationary set of α < κ. This property can be used to show that κ is a limit of most types of large cardinals that are weaker than measurable.\nNotice that the ultrafilter or measure witnessing that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it, which is impossible.\nIf one starts with an elementary embedding j1 of V into M1 with [critical point](https://bliptext.com/articles/critical-point-set-theory) κ, then one can define an ultrafilter U on κ as { S ⊆ κ | κ ∈ j1(S) }.\nThen taking an ultrapower of V over U we can get another elementary embedding j2 of V into M2.\nHowever, it is important to remember that j2 ≠ j1.\nThus other types of large cardinals such as [strong cardinals](https://bliptext.com/articles/strong-cardinal) may also be measurable, but not using the same embedding.\nIt can be shown that a strong cardinal κ is measurable and also has κ-many measurable cardinals below it.\nEvery measurable cardinal κ is a 0-[huge cardinal](https://bliptext.com/articles/huge-cardinal) because κM ⊆ M, that is, every function from κ to M is in M.\nConsequently, Vκ+1 ⊆ M.\n\n# Implications of existence\nIf a measurable cardinal exists, every **Σ**1 2 (with respect to the [analytical hierarchy](https://bliptext.com/articles/analytical-hierarchy)) set of reals has a [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure).\nIn particular, any [non-measurable set](https://bliptext.com/articles/non-measurable-set) of reals must not be **Σ**1 2.\n\n# Real-valued measurable\nA cardinal κ is called **real-valued measurable** if there is a κ-additive [probability measure](https://bliptext.com/articles/probability-measure) on the power set of κ that vanishes on singletons.\nReal-valued measurable cardinals were introduced by.\nshowed that the [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) implies that 𝔠 is not real-valued measurable.\nshowed (see below for parts of Ulam's proof) that real valued measurable cardinals are weakly inaccessible (they are in fact [weakly Mahlo](https://bliptext.com/articles/weakly-mahlo)).\nAll measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ is measurable if and only if κ is greater than 𝔠.\nThus a cardinal is measurable if and only if it is real-valued measurable and strongly inaccessible.\nA real valued measurable cardinal less than or equal to 𝔠 exists if and only if there is a [countably additive](https://bliptext.com/articles/sigma-additivity) extension of the [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) to all sets of real numbers if and only if there is an [atomless](https://bliptext.com/articles/atom-measure-theory) probability measure on the power set of some non-empty set.\nshowed that existence of measurable cardinals in ZFC, real-valued measurable cardinals in ZFC, and measurable cardinals in ZF, are [equiconsistent](https://bliptext.com/articles/equiconsistency).\n\n## Weak inaccessibility of real-valued measurable cardinals\nSay that a cardinal number α is an Ulam number if\nwhenever\nthen\nEquivalently, a cardinal number α is an Ulam number if\nwhenever\nthen\nThe smallest infinite cardinal [ℵ0](https://bliptext.com/articles/aleph-zero) is an Ulam number.\nThe class of Ulam numbers is closed under the [cardinal successor](https://bliptext.com/articles/successor-cardinal) operation.\nIf an infinite cardinal β has an immediate predecessor α that is an Ulam number, assume μ satisfies properties – with X = β. In the [von Neumann model](https://bliptext.com/articles/ordinal-number) of ordinals and cardinals, for each x ∈ β, choose an [injective function](https://bliptext.com/articles/injective-function)\nand define the sets\nSince the fx are one-to-one, the sets\nare pairwise disjoint.\nBy property of μ, the set\nis [countable](https://bliptext.com/articles/countable-set), and hence\nThus there is a b0 such that\nimplying, since α is an Ulam number and using the second definition (with ν = μ and conditions – fulfilled),\nIf b0 < x < β and fx(b0) = ax then x ∈ U(b0, ax).\nThus\nBy property, μ({b0}) = 0, and since |b0| ≤ α, by , and , μ(b0) = 0.\nIt follows that μ(β) = 0.\nThe conclusion is that β is an Ulam number.\nThere is a similar proof that the supremum of a set S of Ulam numbers with |S| an Ulam number is again a Ulam number.\nTogether with the previous result, this implies that a cardinal that is not an Ulam number is [weakly inaccessible](https://bliptext.com/articles/inaccessible-cardinal).\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:52:26.852Z","updated_at":"2025-01-25T18:52:26.852Z","search_vector":"'cardin':2 'measur':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **measurable cardinal** is a certain kind of [large cardinal](https://bliptext.com/articles/large-cardinal) number.\nIn order to define the concept, one introduces a two-valued [measure](https://bliptext.com/articles/measure-mathematics) on a cardinal κ, or more generally on any set.\nFor a cardinal κ, it can be described as a subdivision of all of its [subsets](https://bliptext.com/articles/subset) into large and small sets such that κ itself is large, ∅ and all [singletons](https://bliptext.com/articles/singleton-mathematics) {α} (with α ∈ κ) are small, [complements](https://bliptext.com/articles/set-complement) of small sets are large and vice versa.\nThe [intersection](https://bliptext.com/articles/intersection) of fewer than κ large sets is again large.\nIt turns out that [uncountable](https://bliptext.com/articles/uncountable) cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from [ZFC](https://bliptext.com/articles/zfc).\nThe concept of a measurable cardinal was introduced by [Stanisław Ulam](https://bliptext.com/articles/stanis-aw-ulam) in 1930.\n\n# Definition\nFormally, a measurable cardinal is an uncountable [cardinal number](https://bliptext.com/articles/cardinal-number) κ such that there exists a κ-additive, non-trivial, 0-1-valued [measure](https://bliptext.com/articles/measure-mathematics) μ on the [power set](https://bliptext.com/articles/power-set) of κ.\nHere, κ-additive means: For every λ < κ and every λ-sized set {Aβ}β<λ of pairwise disjoint subsets Aβ ⊆ κ, we have\nEquivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal [ultrafilter](https://bliptext.com/articles/ultrafilter).\nThis means that the intersection of any strictly less than κ-many sets in the ultrafilter, is also in the ultrafilter.\nEquivalently, κ is measurable means that it is the [critical point](https://bliptext.com/articles/critical-point-set-theory) of a non-trivial [elementary embedding](https://bliptext.com/articles/elementary-embedding) of the [universe](https://bliptext.com/articles/universe-set-theory) V into a [transitive class](https://bliptext.com/articles/transitive-class) M.\nThis equivalence is due to [Jerome Keisler](https://bliptext.com/articles/jerome-keisler) and [Dana Scott](https://bliptext.com/articles/dana-scott), and uses the [ultrapower](https://bliptext.com/articles/ultraproduct) construction from [model theory](https://bliptext.com/articles/model-theory).\nSince V is a [proper class](https://bliptext.com/articles/proper-class), a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called [Scott's trick](https://bliptext.com/articles/scott-s-trick).\n\n# Properties\nIt is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be [regular](https://bliptext.com/articles/regular-cardinal).\n(By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2λ.\nIf this were the case, we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1.\nThe intersection of these λ-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure.\nThus, assuming the Axiom of Choice, we can infer that κ is a [strong limit cardinal](https://bliptext.com/articles/strong-limit-cardinal), which completes the proof of its [inaccessibility](https://bliptext.com/articles/inaccessible-cardinal).\nAlthough it follows from [[ZF](https://bliptext.com/articles/zermelo-fraenkel-set-theory)C](https://bliptext.com/articles/zfc) that every measurable cardinal is [inaccessible](https://bliptext.com/articles/inaccessible-cardinal) (and is [ineffable](https://bliptext.com/articles/ineffable-cardinal), [Ramsey](https://bliptext.com/articles/ramsey-cardinal), etc.), it is consistent with [ZF](https://bliptext.com/articles/zermelo-fraenkel-set-theory) that a measurable cardinal can be a [successor cardinal](https://bliptext.com/articles/successor-cardinal).\nIt follows from ZF + [AD](https://bliptext.com/articles/axiom-of-determinacy) that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a [closed and unbounded](https://bliptext.com/articles/club-set) subset.\nUlam showed that the smallest cardinal κ that admits a non-trivial countably-additive two-valued measure must in fact admit a κ-additive measure.\n(If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.\nIf κ is measurable and p ∈ Vκ and M (the ultrapower of V) satisfies ψ(κ, p), then the set of α < κ such that V satisfies ψ(α, p) is stationary in κ (actually a set of measure 1).\nIn particular if ψ is a Π1 formula and V satisfies ψ(κ, p), then M satisfies it and thus V satisfies ψ(α, p) for a stationary set of α < κ. This property can be used to show that κ is a limit of most types of large cardinals that are weaker than measurable.\nNotice that the ultrafilter or measure witnessing that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it, which is impossible.\nIf one starts with an elementary embedding j1 of V into M1 with [critical point](https://bliptext.com/articles/critical-point-set-theory) κ, then one can define an ultrafilter U on κ as { S ⊆ κ | κ ∈ j1(S) }.\nThen taking an ultrapower of V over U we can get another elementary embedding j2 of V into M2.\nHowever, it is important to remember that j2 ≠ j1.\nThus other types of large cardinals such as [strong cardinals](https://bliptext.com/articles/strong-cardinal) may also be measurable, but not using the same embedding.\nIt can be shown that a strong cardinal κ is measurable and also has κ-many measurable cardinals below it.\nEvery measurable cardinal κ is a 0-[huge cardinal](https://bliptext.com/articles/huge-cardinal) because κM ⊆ M, that is, every function from κ to M is in M.\nConsequently, Vκ+1 ⊆ M.\n\n# Implications of existence\nIf a measurable cardinal exists, every **Σ**1 2 (with respect to the [analytical hierarchy](https://bliptext.com/articles/analytical-hierarchy)) set of reals has a [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure).\nIn particular, any [non-measurable set](https://bliptext.com/articles/non-measurable-set) of reals must not be **Σ**1 2.\n\n# Real-valued measurable\nA cardinal κ is called **real-valued measurable** if there is a κ-additive [probability measure](https://bliptext.com/articles/probability-measure) on the power set of κ that vanishes on singletons.\nReal-valued measurable cardinals were introduced by.\nshowed that the [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) implies that 𝔠 is not real-valued measurable.\nshowed (see below for parts of Ulam's proof) that real valued measurable cardinals are weakly inaccessible (they are in fact [weakly Mahlo](https://bliptext.com/articles/weakly-mahlo)).\nAll measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ is measurable if and only if κ is greater than 𝔠.\nThus a cardinal is measurable if and only if it is real-valued measurable and strongly inaccessible.\nA real valued measurable cardinal less than or equal to 𝔠 exists if and only if there is a [countably additive](https://bliptext.com/articles/sigma-additivity) extension of the [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) to all sets of real numbers if and only if there is an [atomless](https://bliptext.com/articles/atom-measure-theory) probability measure on the power set of some non-empty set.\nshowed that existence of measurable cardinals in ZFC, real-valued measurable cardinals in ZFC, and measurable cardinals in ZF, are [equiconsistent](https://bliptext.com/articles/equiconsistency).\n\n## Weak inaccessibility of real-valued measurable cardinals\nSay that a cardinal number α is an Ulam number if\nwhenever\nthen\nEquivalently, a cardinal number α is an Ulam number if\nwhenever\nthen\nThe smallest infinite cardinal [ℵ0](https://bliptext.com/articles/aleph-zero) is an Ulam number.\nThe class of Ulam numbers is closed under the [cardinal successor](https://bliptext.com/articles/successor-cardinal) operation.\nIf an infinite cardinal β has an immediate predecessor α that is an Ulam number, assume μ satisfies properties – with X = β. In the [von Neumann model](https://bliptext.com/articles/ordinal-number) of ordinals and cardinals, for each x ∈ β, choose an [injective function](https://bliptext.com/articles/injective-function)\nand define the sets\nSince the fx are one-to-one, the sets\nare pairwise disjoint.\nBy property of μ, the set\nis [countable](https://bliptext.com/articles/countable-set), and hence\nThus there is a b0 such that\nimplying, since α is an Ulam number and using the second definition (with ν = μ and conditions – fulfilled),\nIf b0 < x < β and fx(b0) = ax then x ∈ U(b0, ax).\nThus\nBy property, μ({b0}) = 0, and since |b0| ≤ α, by , and , μ(b0) = 0.\nIt follows that μ(β) = 0.\nThe conclusion is that β is an Ulam number.\nThere is a similar proof that the supremum of a set S of Ulam numbers with |S| an Ulam number is again a Ulam number.\nTogether with the previous result, this implies that a cardinal that is not an Ulam number is [weakly inaccessible](https://bliptext.com/articles/inaccessible-cardinal).\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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