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Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods

Published online by Cambridge University Press:  12 March 2014

David Pincus
David Pincus*
Affiliation:
University of Washington, Seattle, Washington 98102
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Extract

Fraenkel-Mostowski models are a particularly simple and conceptual tool for proving consistency results involving the axiom of choice, AC. These models satisfy the theory, FM, of a well founded universe of sets built from a ground set of individuals. Zermelo-Fraenkel set theory, ZF, is the extension of FM in which the set of individuals is assumed to be empty. In this paper we show that there is a large class of statements whose consistency with ZF can be proven directly by means of a Fraenkel-Mostowski model.

A statement, Φ, of set theory is said to be transferable if there is a metatheorem: If Φ is true in a Fraenkel-Mostowski model then Φ is consistent with ZF. Jech and Sochor introduced, in [12], the class of boundable statements and proved them to be transferable. Most existential contradictions of AC are boundable. It remains to find criteria under which Ψ ∧ Φ is transferable where Ψ is a universal consequence of AC and Φ is an existential contradiction of AC. To this end we give two classes of statements. Each class is closed under conjunction, contains the boundable statements, and contains a number of universal consequences of AC. Nearly every Fraenkel-Mostowski consistency in the literature falls into one of these two classes.

In §2 we give two generalizations of the boundable statements. In §§3 and 4 the classes of transferable statements are discussed. In §5 we discuss the transfer problem and prove a metatheorem concerning nontransferable statements.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 4 , December 1972 , pp. 721 - 743
Copyright
Copyright © Association for Symbolic Logic 1972

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References

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