Israel Journal of Mathematics

, Volume 48, Issue 2–3, pp 225–243

# Instances of the Conjecture of Chang

• Jean-Pierre Levinski
Article

## Abstract

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω1, and (ρ+++↠(ρ+, ρ), then 0+ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get:

Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K + ≧ω2 and (λ+, λ)↠(K +,K. Then 0+ exists.

Theorem B.Assume that there exists a singularcardinal κ such that(K +,K↠(ω1, ω0. Then 0+ exists.

Theorem C.Assume that (λ ++, λ). Then 0+ exists (also ifK=ω 0.

Remark. Here, as in the paper of Donder and Koepke, “O+ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model.

Theorem D.Assume that (λ ++, λ)↠(K +, K) and thatK≧ω 1. Then 0+ exists.

Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω 2, or that κ is inaccessible. We shall recall this in due place.

Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K +, K↠(ωn +1, ωn), for κ singular andn≧0. 0266 0152 V 3

The consistency of several statements of this sort—a prototype of which is (N ω+1,N ω)↠(ω1, ω0) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.

## Keywords

Regular Cardinal Large Cardinal Measurable Cardinal Unary Predicate Elementary Substructure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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