Editing Inaccessible cardinal (section)

== Models and consistency ==
Suppose that <math>\kappa</math> is a cardinal number. [[Zermelo–Fraenkel set theory]] with Choice (ZFC) implies that the <math>\kappa</math>th level of the [[Von Neumann universe]] <math>V_\kappa</math> is a [[model theory|model]] of ZFC whenever <math>\kappa</math> is strongly inaccessible. Furthermore, ZF implies that the [[Gödel's constructible universe|Gödel universe]] <math>L_\kappa</math> is a model of ZFC whenever <math>\kappa</math> 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 <math>V</math> is a standard model of ZFC and <math>\kappa</math> is an inaccessible in <math>V</math>, then

# <math>V_\kappa</math> is one of the intended models of [[Zermelo–Fraenkel set theory]];
# <math>Def(V_\kappa)</math> is one of the intended models of Mendelson's version of [[Von Neumann–Bernays–Gödel set theory]] which excludes global choice, replacing limitation of size by replacement and ordinary choice;
# and <math>V_{\kappa+1}</math> is one of the intended models of [[Morse–Kelley set theory]]. 

Here, <math>Def(X)</math> is the set of Δ<sub>0</sub>-definable subsets of ''X'' (see [[constructible universe]]). It is worth pointing out that the first claim can be weakened: <math>\kappa</math> does not need to be inaccessible, or even a cardinal number, in order for {{math|<math>V</math><sub><math>\kappa</math></sub>}} to be a standard model of ZF (see [[Inaccessible cardinal#Two model-theoretic characterisations of inaccessibility|below]]).

Suppose <math>V</math> is a model of ZFC. Either <math>V</math> contains no strong inaccessible or, taking <math>\kappa</math> to be the smallest strong inaccessible in <math>V</math>, <math>V_\kappa</math> 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 {{mvar|V}} contains no weak inaccessible or, taking <math>\kappa</math> to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of <math>V</math>, then <math>L_\kappa</math> 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 implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is consistent, 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 if it is consistent.

There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by {{harvtxt|Hrbáček|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'' and preserving powerset of elements of ''M''.