Finitary Set Theory
Laurence Kirby
Source: Notre Dame J. Formal Logic Volume 50, Number 3
(2009), 227-244.
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.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1257862036
Digital Object Identifier: doi:10.1215/00294527-2009-009
Zentralblatt MATH identifier: 05657217
Mathematical Reviews number (MathSciNet): MR2572972
References
[1] Ackermann, W., "Die Widerspruchsfreiheit der allgemeinen Mengenlehre", Mathematische Annalen, vol. 114 (1937), pp. 305--15.
Mathematical Reviews (MathSciNet): MR1513141
Zentralblatt MATH: 0016.19501
Digital Object Identifier: doi:10.1007/BF01594179
[2] Avigad, J., "Saturated models of universal theories", Annals of Pure and Applied Logic, vol. 118 (2002), pp. 219--34.
Mathematical Reviews (MathSciNet): MR1933694
Zentralblatt MATH: 1015.03040
Digital Object Identifier: doi:10.1016/S0168-0072(02)00030-1
[3] Ferreira, F., "A simple proof of Parsons' theorem", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 83--91.
Mathematical Reviews (MathSciNet): MR2131548
Zentralblatt MATH: 1095.03063
Digital Object Identifier: doi:10.1305/ndjfl/1107220675
Project Euclid: euclid.ndjfl/1107220675
[4] Forster, T., Logic, Induction and Sets, vol. 56 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2003.
Mathematical Reviews (MathSciNet): MR1996832
Zentralblatt MATH: 1026.03002
[5] Garcia, N., "Operating on the universe", Archive for Mathematical Logic, vol. 27 (1988), pp. 61--68.
Mathematical Reviews (MathSciNet): MR955312
Zentralblatt MATH: 0633.03045
Digital Object Identifier: doi:10.1007/BF01625835
[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.
Mathematical Reviews (MathSciNet): MR0281602
Zentralblatt MATH: 0215.32601
[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.
Mathematical Reviews (MathSciNet): MR2357524
Zentralblatt MATH: 1137.03019
Digital Object Identifier: doi:10.1305/ndjfl/1193667707
Project Euclid: euclid.ndjfl/1193667707
[9] Kirby, L., "Addition and multiplication of sets", Mathematical Logic Quarterly, vol. 53 (2007), pp. 52--65.
Mathematical Reviews (MathSciNet): MR2288890
Zentralblatt MATH: 1110.03034
Digital Object Identifier: doi:10.1002/malq.200610026
[10] Kirby, L., "A hierarchy of hereditarily finite sets", Archive for Mathematical Logic, vol. 47 (2008), pp. 143--57.
Mathematical Reviews (MathSciNet): MR2410811
Zentralblatt MATH: 1153.03030
Digital Object Identifier: doi:10.1007/s00153-008-0073-7
[11] Lavine, S., "Finite mathematics", Synthese, vol. 103 (1995), pp. 389--420.
Mathematical Reviews (MathSciNet): MR1350265
Zentralblatt MATH: 1059.03523
Digital Object Identifier: doi:10.1007/BF01089734
[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.
Mathematical Reviews (MathSciNet): MR0214464
Zentralblatt MATH: 0265.02027
Digital Object Identifier: doi:10.1007/BF01977295
[13] Montagna, F., and A. Mancini, "A minimal predicative set theory", Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 186--203.
Mathematical Reviews (MathSciNet): MR1295558
Zentralblatt MATH: 0816.03023
Digital Object Identifier: doi:10.1305/ndjfl/1094061860
Project Euclid: euclid.ndjfl/1094061860
[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.
Mathematical Reviews (MathSciNet): MR519815
Zentralblatt MATH: 0442.03042
[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.
Mathematical Reviews (MathSciNet): MR0280330
Zentralblatt MATH: 0202.01202
Digital Object Identifier: doi:10.1016/S0049-237X(08)70771-7
[16] Parsons, C., "On $n$"-quantifier induction, The Journal of Symbolic Logic, vol. 37 (1972), pp. 466--82.
Mathematical Reviews (MathSciNet): MR0325365
Zentralblatt MATH: 0264.02027
Digital Object Identifier: doi:10.2307/2272731
JSTOR: links.jstor.org
[17] Previale, F., "Induction and foundation in the theory of hereditarily finite sets", Archive for Mathematical Logic, vol. 33 (1994), pp. 213--41.
Mathematical Reviews (MathSciNet): MR1278334
Zentralblatt MATH: 0810.03048
Digital Object Identifier: doi:10.1007/BF01203033
[18] Rödding, D., "Primitiv-rekursive Funktionen über einem Bereich endlicher Mengen", Archiv für mathematische Logik und Grundlagenforschung, vol. 10 (1967), pp. 13--29.
Mathematical Reviews (MathSciNet): MR0214463
Zentralblatt MATH: 0189.00901
[19] Świerczkowski, S., "Finite sets and Gödel's incompleteness theorems", Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 422 (2003), p. 58.
Mathematical Reviews (MathSciNet): MR2030226
Zentralblatt MATH: 1058.03065
[20] Tait, W. W., "Finitism", Journal of Philosophy, vol. 78 (1981), pp. 524--46.
Zentralblatt MATH: 1024.03008
[21] Tarski, A., "Sur les ensembles finis", Fundamenta Mathematicae, vol. 6 (1924), pp. 45--95.
Zentralblatt MATH: 50.0135.02
[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.
Mathematical Reviews (MathSciNet): MR1015503
Zentralblatt MATH: 0606.01026
[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.
Mathematical Reviews (MathSciNet): MR920815
Zentralblatt MATH: 0654.03036
[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.
Mathematical Reviews (MathSciNet): MR0209111
Zentralblatt MATH: 0183.00601
[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.
Mathematical Reviews (MathSciNet): MR0060441
Zentralblatt MATH: 0051.24602
Digital Object Identifier: doi:10.1007/BF01343173
[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.
Zentralblatt MATH: 40.0098.03
Notre Dame Journal of Formal Logic
