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Support structures for the axiom of choice

  • David Pincus (a1)

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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.

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[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. 83134.
[2]Jech, T. and Sochor, A., On θ model of the set theory, Bull. Acad. Polonaise Sci. Sec. Sci. Math., Astr. Phy., 14 (1966).
[3]Levy, A., The interdependence of certain consequences of the axiom of choice, Fundamenta Mathematicae, vol. 54 (1964), pp. 135157.
[4]Mostowski, A., Über die Unabhängigkeit des Wohlordnungssatz vom Ordnungsprinzip, Fundamenta Mathematicae, vol. 32 (1939), pp. 201252.
[5]Pincus, D., Zermelo-Fraenkel consistencies by Frankel-Mostowski methods, to appear in this Journal.
[6]Shoenfield, J. R., Unramified forcing, Proceedings of the 1967 Summer Institute on Set Theory, pp. 357381.
[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.

Support structures for the axiom of choice

  • David Pincus (a1)

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