Inaccessible Cardinal - Two Model-theoretic Characterisations of Inaccessibility

Two Model-theoretic Characterisations of Inaccessibility

Firstly, a cardinal κ is inaccessible if and only if κ has the following reflection property: for all subsets U ⊂ V_κ, there exists α < κ such that is an elementary substructure of . (In fact, the set of such α is closed unbounded in κ.) Equivalently, κ is -indescribable for all n ≥ 0.

It is provable in ZF that ∞ satisfies a somewhat weaker reflection property, where the substructure (V_α, ∈, U ∩ V_α) is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation can be defined, truth itself cannot, due to Tarski's theorem.

Secondly, under ZFC it can be shown that κ is inaccessible if and only if (V_κ, ∈) is a model of second order ZFC.

In this case, by the reflection property above, there exists α < κ such that (V_α, ∈) is a standard model of (first order) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a standard model of ZFC.