Chang’s Conjecture, Generic Elementary Embeddings and Inner Models for Huge Cardinals

Abstract

We introduce a natural principle Strong Chang Reflection strengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show that decisive ideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10]. 1,2 Much of 20th century logic in general and model theory in particular was tied up with understanding the expressive power of first and second order logic (and their variants). Of particular interest is the role of the downwards Lowenheim-Skolem theorem; indeed in some ways it is the distinguishing feature of first order logic ([19]).The Downward Lowenheim-Skolem theorem states that if A is a structure in a countable language then for all infinite cardinals κ less than the cardinality of A there is a B ≺ A of cardinality κ. Tremendous effort was put into generalizing the downwards Lowenheim-Skolem theorem in an attempt to show that elementary substructure B can be taken to have some second order properties.The coarsest second order properties have to do with cardinality. In this paper we consider various more subtle second order properties. Among them is being correct for the nonstationary ideal. By demanding that B have these properties we are able to formulate a version of the downwards Lowenheim-Skolem theorem with very strong large cardinal strength. 3 §1. Two cardinal transfer theorems. Let L be a countable language with a distinguished unary predicate R. Then R j , c k . . . i,j,k∈ is said to have type (κ, ) if and only if |A| = κ and |R A | = .

In Memoriam: James Earl Baumgartner (1943-2011)

“…Foreman [95] highlighted the power of non-stationary ideals and their restrictions to selected stationary sets when he showed that the consistency of ZFC together with his strengthening of the classical Chang conjectures to the principle of Strong Chang Reflection 36 for (ω n+3 , ω n ) implies the consistency of ZFC together with the existence of a huge cardinal in a model of the form L[A * , Ȋ] where Ȋ is the dual of the appropriate nonstationary ideal. Foreman used the proposition below to show that the set A * was absolutely definable.…”