Regular cardinals in models of ${\rm ZF}$

Abstract

Abstract. We prove the following Theorem. Suppose M is a countable model of ZFC and k is an almost huge cardinal in M. Let A be a subset of k consisting of nonlimit ordinals. Then there is a model NA of ZF such that S0 is a regular cardinal in NA iff a e A for every a > 0. 0. Introduction. We consider the following question. What are the restrictions in ZF on the class of all regular cardinals? Clearly, S0 is always regular and Nu, Sw+l0, SuU and Sa, for a the least s.t. Sa = a, are singular. For a limit a, if Na is regular, then it is already quite large and its existence is unprovable in ZF. In ZFC, Na+1 is a regular cardinal for every a. Feferman and Levy [3] proved that it is not true in ZF alone. They built a model of ZF + "Sj is singular". Combining their method with Easton's work [2], it is possible to build a model of ZF + "for every a, Xa or Sa + 1 is singular". Now to make both S" and Sa+1 some additional assumptions are needed.Thus, the Dodd and Jensen Covering Lemma [1] implies that at least O1' is needed, and by Mitchell [11] it looks like at least a hypermeasurable cardinal is needed. On the other hand, a model of ZF + "Va > 0 Na is a singular cardinal" is constructed in [5] from the much stronger assumption Con(ZFC + "V«3k > a k is strongly compact"). The simplest question, which is unclear how to handle by methods of [5], is the following. Is there a model of ZF s.t. those regular alephs are only N0 and Hx! Or, for example, let A = {a + I20\a g On}. Is there a model of ZF s.t. those regular alephs are exactly {Sa\a G A U {0}}, i.e., N0,N120,N240,...?We shall prove the following Theorem. Suppose M is a countable model of ZFC and k is an almost huge cardinal in M. Let A be a subset of k consisting of nonlimit ordinals. Then there is a model NA of ZF s.t. those regular alephs are exactly {NJa G A U {0}}.As an immediate consequence we obtain Theorem. If ZFC + (3k) (k is an almost huge cardinal) is consistent, then there is a model M of ZFC s.t. for every class A of M consisting of nonlimit ordinals there exists a model NA ofZFs.t.those regular alephs are exactly {NJa E^U {0}}.Notice, that for A = 0 it gives a model with all uncountable Sa's singular, but the assumption here is stronger than those of [5].