Journal of Symbolic Logic

Extended Ultrapowers and the Vopenka-Hrbacek Theorem without Choice

Mitchell Spector

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We generalize the ultrapower in a way suitable for choiceless set theory. Given an ultrafilter, forcing is used to construct an extended ultrapower of the universe, designed so that the fundamental theorem of ultrapowers holds even in the absence of the axiom of choice. If, in addition, we assume DC, then an extended ultrapower of the universe by a countably complete ultrafilter must be well-founded. As an application, we prove the Vopenka-Hrbacek theorem from ZF + DC only (the proof of Vopenka and Hrbacek used the full axiom of choice): if there exists a strongly compact cardinal, then the universe is not constructible from a set. The same method shows that, in $\mathrm{L}\lbrack 2^\omega\rbrack$, there cannot exist a $\theta$-compact cardinal less than $\theta$ (where $\theta$ is the least cardinal onto which the continuum cannot be mapped); a similar result can be proven for other models of the form $\mathrm{L}\lbrack A \rbrack$. The result for $\mathrm{L}\lbrack 2^\omega\rbrack$ is of particular interest in connection with the axiom of determinacy. The extended ultrapower construction of this paper is an improved version of the author's earlier pseudo-ultrapower method, making use of forcing rather than the omitting types theorem.

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J. Symbolic Logic, Volume 56, Issue 2 (1991), 592-607.

First available in Project Euclid: 6 July 2007

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Spector, Mitchell. Extended Ultrapowers and the Vopenka-Hrbacek Theorem without Choice. J. Symbolic Logic 56 (1991), no. 2, 592--607.

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