this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible. It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2λ. If this were the case, then we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1. The intersection of these λ-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure. Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility. If κ is measurable and p∈Vκ and M (the ultrapower of V) satisfies ψ(κ,p), then the set of α < κ such that V satisfies ψ(α,p) is stationary in κ (actually a set of measure 1). In particular if ψ is a Π1 formula and V satisfies ψ(κ,p), then M satisfies it and thus V satisfies ψ(α,p) for a stationary set of α < κ. This property can be used to show that κ is a limit of most types of large cardinals that are weaker than measurable. Notice that the ultrafilter or measure witnessing that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it, which is impossible. If one starts with an elementary embedding j1 of V into M1 with critical point κ, then one can define an ultrafilter U on κ as { S⊆κ : κ∈j1(S) }. Then taking an ultrapower of V over U we can get another elementary embedding j2 of V into M2. However, it is important to remember that j2 ≠ j1. Thus other types of large cardinals such as strong cardinals may also be measurable,