On the Singular Cardinals Problem

Abstract

In this paper we show, for example, that if the GCH holds for every cardinal less than tc, a singular cardinal of uncountable cofinality, then the GCH holds at tc itself. This result is contrary to the previous expectations of nearly all set-theorists, including myself. Another consequence of Theorem 1.1 is that if the GCH holds for every singular cardinal cofinal with œ9 then it holds for every singular cardinal. The immediate stimulus for this result was some work of Kanamori and Magidor concerning nonregular uniform ultrafilters over œ\. The other principal influences were a result of Scott concerning the GCH at measurable cardinals, some work of Keisler on ultrapowers of the sort defined in 1.3, the two-cardinal theory developed by several model-theorists, some work of Prikry and Silver on indecomposable ultrafilters [3], [4], as well as Cohen's methods and work on nonstandard models of set theory [2]. Our terminology is mostly standard. If £ is a cardinal, S is called a stationary subset of tc if it intersects every closed cofinal subset of tc. A function h : X -> tc is continuous if, for every limit ordinal a e X9 h(a) is the least upper bound of {h(s) : s e a}. If tc is a cardinal, /c is the least cardinal greater than tc. Also, KW is the sth cardinal greater than tc. Thus tc = tc, tc = tc9 etc. The cofinality of tc is X iff X is the least cardinal such that tc can be written as a union of X sets, each of cardinality < tc. tc is singular iff its cofinality is < tc.