Instances of the Conjecture of Chang
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+, 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.
KeywordsRegular Cardinal Large Cardinal Measurable Cardinal Unary Predicate Elementary Substructure
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