Ramsey cardinals, α-Erdös cardinals, and the core model

Abstract

The core model K was introduced by R. B. Jensen and A. J. Dodd [DoJ]. K is the union of Gödel's constructible universe L together with all mice, i.e., , and K is a transitive model of ZFC + (V = K) + GCH (see [DoJ]). V = K is consistent with the existence of Ramsey cardinals [M], and if cf(α) > ω, V = K is consistent with the existence of α-Erdös cardinals [J]. Let K be Ramsey. Then there is a smallest inner model Wκ of ZFC in which κ is Ramsey. We have Wκ ⊨ V = K and Wκ ⊆ K [M]. The existence of Wκ with is equivalent to the existence of a sharplike mouse on N ⊨ K with N ⊨ κ Ramsey. (A mouse N on is called sharplike provided .) We have , where is the mouse iteration of N. N is the oleast mouse not in Wκ (see [J] and [DJKo]). Here < denotes the mouse order. The context always clarifies whether the mouse order or the usual <-relation is meant.The main result of §1 is that Wκ ⊨ κ is the only Ramsey cardinal. A similar result has been found true in the smallest inner model L[U] of ZFC + “κ is measurable” if U is a normal measure on κ: L[U] ⊨ κ is the only measurable cardinal [Ku].