Journal of Symbolic Logic

Jonsson-like partition relations and j: V → V

Arthur W. Apter and Grigor Sargsyan

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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 μ → [μ]ω2. 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.

Article information

J. Symbolic Logic, Volume 69, Issue 4 (2004), 1267-1281.

First available in Project Euclid: 2 December 2004

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Zentralblatt MATH identifier

Primary: 03E02: Partition relations 03E35: Consistency and independence results 03E55: Large cardinals 03E65: Other hypotheses and axioms

Jonsson cardinals partition relations polarized partitions elementary embeddings


Apter, ArthurW.; Sargsyan, Grigor. Jonsson-like partition relations and j: V → V. J. Symbolic Logic 69 (2004), no. 4, 1267--1281. doi:10.2178/jsl/1102022223.

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