Inaccessible Cardinal - Models and Consistency

Models and Consistency

ZFC implies that the V_κ is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that the Gödel universe L_κ is a model of ZFC whenever κ is weakly inaccessible. Thus ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal.

If V is a standard model of ZFC and κ is an inaccessible in V, then: V_κ is one of the intended models of Zermelo–Fraenkel set theory; and Def (V_κ) is one of the intended models of Von Neumann–Bernays–Gödel set theory; and V_κ+1 is one of the intended models of Morse–Kelley set theory. Here Def (X) is the Δ₀ definable subsets of X (see constructible universe). However, κ does not need to be inaccessible, or even a cardinal number, in order for V_κ to be a standard model of ZF (see below).

Suppose V is a model of ZFC. Either V contains no strong inaccessible or, taking κ to be the smallest strong inaccessible in V, V_κ is a standard model of ZFC which contains no strong inaccessibles. Thus, the consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then L_κ is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles". This shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals.

The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched in the previous paragraph that the consistency of ZFC + "there is an inaccessible cardinal" implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible cardinal" can be formalized in ZFC. This follows from Gödel's second incompleteness theorem, which shows that if ZFC + "there is an inaccessible cardinal" is consistent, then it cannot prove its own consistency. Because ZFC + "there is an inaccessible cardinal" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + "there is an inaccessible cardinal" then this latter theory would be able to prove its own consistency, which is impossible.

There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by Hrbacek & Jech (1999, p. 279), is that the class of all ordinals of a particular model M of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending M.