Indescribable cardinal

rdf:langString Indescribable cardinal

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rdf:langString In mathematics, a Q-indescribable 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 . A cardinal number κ is called Πnm-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ.Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. Σnm-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+1-th order logic with m-1 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 Πnm-indescribable for all positive integers m and n. If α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of Vκ such that φ(U) holds in Vκ+α there is a some λ<κ such that φ(U ∩ Vλ) holds in Vλ+α. If α is infinite then α-indescribable ordinals are totally indescribable, and if α is finite they are the same as Παω-indescribable ordinals. α-indescribability implies that α<κ, but there is an alternative notion of shrewd cardinals that makes sense when α≥κ: there is λ<κ and β such that φ(U ∩ Vλ) holds in Vλ+β.