On successors of Jónsson cardinals

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero 4] calculates correctly the successors of JJ onsson cardinals, assuming O Sword does not exist. Namely, if is a JJ onsson cardinal then + = +K , provided that there is no non-trivial elementary embedding j : K ?! K. There are a number of related results in ZF C concerning P() in V and inner models, for a JJ onsson or singular cardinal. x1 Introduction An inner core model, built under some assumption on the limitation of size of the universe V of sets, reeects many of the cardinality and coonality properties of V. Such \Covering Properties" are well known, and such results of Jensen's, which assert that if O # does not exist, then singular implies +V = +L , and from which also follows the fact that weakly compact implies +V = +L , are now seen as precursors of a family of theorems related to generalisations of L-the so-called core models, also built under some assumption that will imply the model's rigidity. (By a model M's rigidity we mean that there is no non-trivial elementary embedding j : M ?! M.) With the Dodd-Jensen core model (which can be viewed as being built under the assumption there is no inner model of a measurable cardinal) the possibility was open for further properties to reeect from V to the \generalised constructible hierarchy" of K DJ : namely Erd} os, JJ onsson, and Ramsey properties. Recall that a cardinal is JJ onsson if every algebra A = hA; (f n) n<! i; A (where f n is a sequence of nitary functions) has a proper elementary subalgebra A 0 of cardinality. (See 3]). Ramsey, and JJ onsson cardinals are Ramsey (and so JJ onsson) in K DJ by results of Jensen 1] and Mitchell 5]. Ramsey cardinals are already weakly compact (see Jech 2]) and it was known that weakly compact implied +V = +K DJ. We denote here by K the core model for measures of order zero, (see 4]), where measures are allowed, as long as they have no measure 1 set concentrating on smaller measurables. It is known (under the working assumption that K is rigid, which we abbreviate as \ O sword does not exist" or \:O s "), that the successors of singulars are …