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Jonsson-like partition relations and j: V → V

Published online by Cambridge University Press:  12 March 2014

Arthur W. Apter  and
Grigor Sargsyan
Arthur W. Apter
Affiliation:
Department of Mathematics, Baruch College of CUNY, New York 10010, USA, E-mail: [email protected]
Grigor Sargsyan
Affiliation:
Group in Logic and The Methodology of Science, University of CaliforniaBerkeley, California 94720, USA, E-mail: [email protected]
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Abstract.

Working in the theory ”ZF + There is a nontrivial elementary embedding j : V → V“, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation . We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j: V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.

Keywords

Jonsson cardinalspartition relationspolarized partitionselementary embeddings
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 4 , December 2004 , pp. 1267 - 1281
Copyright
Copyright © Association for Symbolic Logic 2004

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