Strongly compact cardinals, elementary embeddings and fixed points

Extract

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on P_κλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.

We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ^<κ is defined to be sup_γ<κγ^γ. Cardinal exponentiation is always associated from the top. Thus, for example, 2^λ<κ means 2^(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣^M and P(X)^M denote the cardinality of X in M and the power set of X in M respectively.

We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)

Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, P_κλ = {X ⊂ λ∣∣X∣ < κ}.

It is important to note that ∣P_κλ∣ = λ^{< κ}.