Notre Dame Journal of Formal Logic

Finitary Set Theory

Laurence Kirby

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1257862036

Digital Object Identifier
doi:10.1215/00294527-2009-009

Mathematical Reviews number (MathSciNet)
MR2572972

Zentralblatt MATH identifier
1190.03043

Subjects
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

Keywords
hereditarily finite sets primitive recursive set functions adjunction

Citation

Kirby, Laurence. Finitary Set Theory. Notre Dame J. Formal Logic 50 (2009), no. 3, 227--244. doi:10.1215/00294527-2009-009. http://projecteuclid.org/euclid.ndjfl/1257862036.


Export citation

References

  • [1] Ackermann, W., "Die Widerspruchsfreiheit der allgemeinen Mengenlehre", Mathematische Annalen, vol. 114 (1937), pp. 305--15.
  • [2] Avigad, J., "Saturated models of universal theories", Annals of Pure and Applied Logic, vol. 118 (2002), pp. 219--34.
  • [3] Ferreira, F., "A simple proof of Parsons' theorem", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 83--91.
  • [4] Forster, T., Logic, Induction and Sets, vol. 56 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2003.
  • [5] Garcia, N., "Operating on the universe", Archive for Mathematical Logic, vol. 27 (1988), pp. 61--68.
  • [6] Givant, S., and A. Tarski, "Peano arithmetic and the Zermelo-like theory of sets with finite ranks", Notices of the American Mathematical Society, vol. 77T-E51 (1977), pp. A--437.
  • [7] Jensen, R. B., and C. Karp, "Primitive recursive set functions", pp. 143--76 in Axiomatic Set Theory (Proceedings of the Symposia in Pure Mathematics, Vol. XIII, Part I, University of California, Los Angeles, 1967), American Mathematical Society, Providence, 1971.
  • [8] Kaye, R., and T. L. Wong, "On interpretations of arithmetic and set theory", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 497--510.
  • [9] Kirby, L., "Addition and multiplication of sets", Mathematical Logic Quarterly, vol. 53 (2007), pp. 52--65.
  • [10] Kirby, L., "A hierarchy of hereditarily finite sets", Archive for Mathematical Logic, vol. 47 (2008), pp. 143--57.
  • [11] Lavine, S., "Finite mathematics", Synthese, vol. 103 (1995), pp. 389--420.
  • [12] Mahn, F.-K., "Zu den primitiv-rekursiven Funktionen über einem Bereich endlicher Mengen", Archiv für mathematische Logik und Grundlagenforschung, vol. 10 (1967), pp. 30--33.
  • [13] Montagna, F., and A. Mancini, "A minimal predicative set theory", Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 186--203.
  • [14] Paris, J. B., and L. A. S. Kirby, "$\Sigma \sbn$"-collection schemas in arithmetic", pp. 199--209 in Logic Colloquium '77 (Proceedings of the Conference, Wrocław, 1977), vol. 96 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1978.
  • [15] Parsons, C., "On a number theoretic choice schema and its relation to induction", pp. 459--73 in Intuitionism and Proof Theory (Proceedings of the Conference, Buffalo, 1968), North-Holland, Amsterdam, 1970.
  • [16] Parsons, C., "On $n$"-quantifier induction, The Journal of Symbolic Logic, vol. 37 (1972), pp. 466--82.
  • [17] Previale, F., "Induction and foundation in the theory of hereditarily finite sets", Archive for Mathematical Logic, vol. 33 (1994), pp. 213--41.
  • [18] Rödding, D., "Primitiv-rekursive Funktionen über einem Bereich endlicher Mengen", Archiv für mathematische Logik und Grundlagenforschung, vol. 10 (1967), pp. 13--29.
  • [19] Świerczkowski, S., "Finite sets and Gödel's incompleteness theorems", Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 422 (2003), p. 58.
  • [20] Tait, W. W., "Finitism", Journal of Philosophy, vol. 78 (1981), pp. 524--46.
  • [21] Tarski, A., "Sur les ensembles finis", Fundamenta Mathematicae, vol. 6 (1924), pp. 45--95.
  • [22] Tarski, A., The notion of rank in axiomatic set theory and some of its applications, edited by S. R. Givant and R. N. McKenzie, Contemporary Mathematicians. Birkhäuser Verlag, Basel, 1986. Originally published in Bulletin of the American Mathematical Society, vol. 61 (1955), Abstract 628, p. 443.
  • [23] Tarski, A., and S. Givant, A Formalization of Set Theory without Variables, vol. 41 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1987.
  • [24] von Neumann, J., "An axiomatization of set theory", pp. 393--413 in From Frege to Gödel. A Source Book in Mathematical Logic, 1879--1931, edited by J. van Heijenoort, Harvard University Press, Cambridge, 1967. Translation of ``Eine axiomatisierung der Mengenlehre,'' Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219--40.
  • [25] Wang, H., "Between number theory and set theory", Mathematische Annalen, vol. 126 (1953), pp. 385--409. Reprinted in Logic, Computers, and Sets, Chelsea Publishing Co., New York, 1970, pp. 478--506.
  • [26] Whitehead, A. N., and B. Russell, Principia Mathematica, 2d edition, Cambridge University Press, Cambridge, 1963.
  • [27] Zermelo, E., "Sur les ensembles finis et le principe de l'induction complète", Acta Mathematica, vol. 32 (1909), pp. 185--93.