ⓘ Indescribable cardinal. In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to describe in some language Q. There ..

ⓘ Indescribable cardinal

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 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 m-1 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+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 Π n m -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 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.

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