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Support structures for the axiom of choice

Published online by Cambridge University Press:  12 March 2014

David Pincus
David Pincus*
Affiliation:
University of Washington, Seattle, Washington 98105
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Extract

The notion of “support” was introduced by Mostowski in [4] in order to prove that a certain universe satisfied the ordering principle but not the axiom of choice. The notion was refined in [3] and in [1] it was shown to be satisfied in a certain Cohen model of full ZF set theory. This paper is an axiomatic study of universes whose undefined relations are ∈ and a “support structure”, T.

In §2 the general theory is introduced and the universes of [4] and [1] are characterized. §3 examines a more complicated universe which will be used in [5] to show that in many cases a consistency in full ZF set theory may be proven directly by the methods of [4]. The embedding theorems of §4 are crucial to this application.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 1 , March 1971 , pp. 28 - 38
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Halpern, J. D. and Levy, A., The Boolean prime ideal theorem does not imply the axiom of choice, Proceedings of the 1967 UCLA Summer Institute on Set Theory, pp. 83–134.Google Scholar
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[5]Pincus, D., Zermelo-Fraenkel consistencies by Frankel-Mostowski methods, to appear in this Journal.Google Scholar
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[7]Solovay, R. M. and Scott, D., Boolean valued models of set theory, Proceedings of the 1967 Summer Institute on Set Theory, lecture notes, University of California, Berkeley, California, U.S.A.Google Scholar