Notre Dame Journal of Formal Logic

Finitary Set Theory

Laurence Kirby


I argue for the use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory. It allows a simplified axiomatization for the first-order theory of hereditarily finite sets based on an induction schema and a rigorous characterization of the primitive recursive set functions. The latter leads to a primitive recursive presentation of arithmetical operations on finite sets.

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Notre Dame J. Formal Logic, Volume 50, Number 3 (2009), 227-244.

First available in Project Euclid: 13 November 2009

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Primary: 03C13: Finite structures [See also 68Q15, 68Q19] 03D20: Recursive functions and relations, subrecursive hierarchies 03E10: Ordinal and cardinal numbers 03E30: Axiomatics of classical set theory and its fragments

hereditarily finite sets primitive recursive set functions adjunction


Kirby, Laurence. Finitary Set Theory. Notre Dame J. Formal Logic 50 (2009), no. 3, 227--244. doi:10.1215/00294527-2009-009.

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