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 Hanf & Scott (1961).
[1] Following Lévy's hierarchy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.
-indescribable cardinals are defined in a similar way, but with an outermost existential quantifier.
Prior to defining the structure
, one new predicate symbol is added to the language of set theory, which is interpreted as
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.
[citation needed] The cardinal number
is called totally indescribable if it is
is an ordinal, the cardinal number
-indescribable ordinals are totally indescribable, and if
, but there is an alternative notion of shrewd cardinals that makes sense when
9, theorem 4.3 Originally, a cardinal κ was called Q-indescribable if for every Q-formula
satisfying the above version based on the cumulative hierarchy were called strongly Q-indescribable.
[6] This property has also been referred to as "ordinal
[8] A cardinal is inaccessible if and only if it is
59 The indescribability condition is equivalent to
satisfying the reflection principle (which is provable in ZFC), but extended by allowing higher-order formulae with a second-order free variable.
[10], Corollary 4.3 If V=L, then for a natural number n>0, an uncountable cardinal is Π1n-indescribable iff it's (n+1)-stationary.
[1]p.277 This gives a tool to show necessary properties of indescribable cardinals.
-indescribable and the set of such cardinals below it is stationary.
-indescribable cardinal (this is proved from consistency of ZFC with
[citation needed] Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for
denotes elementary equivalence.
this is a biconditional (see Two model-theoretic characterisations of inaccessibility).
Measurable cardinals are
-indescribable, but the smallest measurable cardinal is not
61 However, assuming choice, there are many totally indescribable cardinals below any measurable cardinal.