## 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 ≧ ω_{1}, 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* ^{+} ≧ω_{2} and (λ^{+}, λ)↠(*K* ^{+},K. Then 0^{+} exists.

Theorem B.*Assume that there exists a* singular*cardinal κ such that*(*K* ^{+},K↠(ω_{1}, ω_{0}. Then 0^{+} exists.

Theorem C.*Assume that (λ* ^{++}, λ). Then 0^{+} exists (also if*K=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 over*L*(μ), but the interest is in expecting more cardinals, coming from a larger core model.

Theorem D.*Assume that (λ* ^{++}, λ)↠(*K* ^{+}, K) and that*K≧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 that*K≧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↠(ω_{n} +1, ω_{n}), for κ singular and*n*≧0. 0266 0152 V 3

The consistency of several statements of this sort—a prototype of which is (**N** _{ω+1},**N** _{ω})↠(ω_{1}, ω_{0}) —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## Preview

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