ⓘ Indescribable cardinal
In mathematics, a Qindescribable cardinal is a certain kind of large cardinal number that is hard to describe in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by Hanf & Scott.
A cardinal number κ is called Π n m indescribable if for every Π m proposition φ, and set A ⊆ V κ with V κ+n, ∈, A ⊧ φ there exists an α < κ with V α+n, ∈, A ∩ V α ⊧ φ. Here one looks at formulas with m1 alternations of quantifiers with the outermost quantifier being universal. Σ n m indescribable cardinals are defined in a similar way. The idea is that κ cannot be distinguished looking from below from smaller cardinals by any formula of n+1th order logic with m1 alternations of quantifiers even with the advantage of an extra unary predicate symbol for A. This implies that it is large because it means that there must be many smaller cardinals with similar properties.
The cardinal number κ is called totally indescribable if it is Π n m indescribable for all positive integers m and n.
If α is an ordinal, the cardinal number κ is called αindescribable if for every formula φ and every subset U of V κ such that φU holds in V κ+α there is a some λ 1 it is consistent with ZFC that the least Σ m n indescribable exists and is above the least Π m n indescribable cardinal this is proved from consistency of ZFC with Π m n indescribable cardinal and a Σ m n indescribable cardinal above it.
Measurable cardinals are Π 2 1 indescribable, but the smallest measurable cardinal is not Σ 2 1 indescribable. However there are many totally indescribable cardinals below any measurable cardinal.
Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for Π m n and Σ m n indescribability.
 Kentucky Cup Distaff Stakes Indescribable cardinal a term used in mathematics to describe a certain kind of large cardinal number that is hard to describe
 shrewd cardinal is a certain kind of large cardinal number introduced by Rathjen 1995 , extending the definition of indescribable cardinals A cardinal number
 A cardinal k is κ  strongly unfoldable, and κ  unfoldable, if and only if it is weakly compact. A κ ω  unfoldable cardinal is totally indescribable and
 strongly unfoldable and therefore totally indescribable Kanamori, Akihiro 2003 The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings
 cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ
 2010  Askbut I Wonttell 4 2009  Acoma 4 Jesus Castanon 2008  Indescribable 4 Kent Desormeaux 2007  Criminologist 4 John Velazquez 2006
 Subtle cardinal Supercompact cardinal Superstrong cardinal Totally indescribable cardinal Weakly compact cardinal Weakly hyper  Woodin cardinal Weakly
 August Hlond July 5, 1881 October 22, 1948 was a Polish cardinal of the Roman Catholic Church, who was Archbishop of Poznan and Gniezno in 1926 and
 elements are comparable totally indescribable A totally indescribable cardinal is a cardinal that is Πm n  indescribable for all m, n transfinite 1. An
 soprano, with a limpid quality that musicians of the time declared was indescribable His intonation was perfect and all musicians spoke of his length of

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