In set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by .

is called -indescribable. Following Lévy's hierarchy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. -indescribable. 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.

is called totally indescribable-indescribable for all positive integers m and n.p. 59

is called -indescribable, but there is an alternative notion of shrewd cardinals..Ch. 9, theorem 4.3

Historical note

..p.207 satisfying the above version based on the cumulative hierarchy were called strongly Q-indescribable.-indescribability".p.32

Equivalent conditions

-indescribable.p. 59 A cardinal is inaccessible-indescribable.

-indescribable cardinals are the same as weakly compact cardinals.p. 59

satisfying the reflection principle (which is provable in ZFC), but extended by allowing higher-order formulae with a second-order free variable.

a small embedding., Corollary 4.3

If V=L, then for a natural number n>0, an uncountable cardinal is Π-indescribable iff it's (n+1)-stationary.

Enforceable classes

hold.p.277 This gives a tool to show necessary properties of indescribable cardinals.

Properties

-indescribable..-indescribable and the set of such cardinals below it is stationary. The consistency strength-indescribable cardinal above it).

Totally indescribable cardinals remain totally indescribable in the constructible universep. 62--63-indescribability.

denotes elementary equivalence. this is a biconditional (see Two model-theoretic characterisations of inaccessibility).

-indescribable, but the smallest measurable cardinal-indescribable.p. 61 However, assuming choice, there are many totally indescribable cardinals below any measurable cardinal.

-indescribable cardinal".

References

Citations