By the elementarity of j, V must think the same of the preimage of j(κ), namely κ — so V thinks there is a strongly inaccessible cardinal below κ

It is easy to see that for any such embedding Vκ+1⊆ M where κ = crit(j). This amount of agreement enables one to show that κ is strongly inaccessible, Mahlo, indescribable, ineffable, etc. To illustrate this let us assume that we have shown that κ is strongly inaccessible and let us show that κ has much stronger large cardinal properties. Since κ is strongly inaccessible in V and since (Vκ+1)M =Vκ+1, M also thinks that κ is strongly inaccessible. In particular, M thinks that there is a stron