Ramsey cardinals and constructibility

Extract

Large cardinal properties divide rather strikingly into two groups. “Small” large cardinal properties such as weak compactness always relativize to L, while in contrast “large” large cardinal properties such as Ramsey are incompatible with L. These properties seem to be similar otherwise and this sense of similarity is reinforced by the fact that many of the large cardinals do exist in the L-like model L(μ). This paper will show that the division is caused by an artificially restrictive class of “constructible” sets rather than an essential difference in the properties themselves. Specifically, we consider the class K of sets constructible from mice as defined by Dodd and Jensen [3] and prove

Theorem 1. If ρ is Ramsey then ρ is Ramsey in K.

A modification of the proof will show

Theorem 2. If ρ is Jonson, then ρ is Ramsey in K.

Since Ramsey implies Jonson this shows that the notions are equiconsistent. Theorem 2 was proved by Kunen [5] under the assumption that V = L(μ).

The proof of Theorems 1 and 2 depends heavily on results of Dodd and Jensen [3] about K and mice. These results are stated without proof in §2. §2 also contains elementary (to a reader familiar with the theory of iterated ultrapowers) proofs of special cases of some of these lemmas sufficient to give a self contained proof of the following corollary of Theorem 1:

Corollary. If ρ ≤ κ,ρ is Ramsey and L(μ) ⊨ μ is a measure on κ then L(μ) ⊨ ρ is Ramsey.