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  • Print publication year: 2011
  • Online publication date: October 2011

ω-models of finite set theory


Abstract. Finite set theory, here denoted ZFfin, is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZFfin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In particular, we present a new method for building ω-models of ZFfin that leads to a perspicuous construction of recursive nonstandard ω-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an ω-model. The central theorem of the paper is the following:

Theorem A. For every graph (A, F), where F is a set of unordered pairs of A, there is an ω-model m of ZFfin whose universe contains A and which satisfies the following conditions:

(1) (A, F) is definable in m;

(2) Every element of m is definable in (m, a)aA;

(3) If (A, F) is pointwise definable, then so is m;

(4) Aut(m) ≅ Aut(A, F).

Theorem A enables us to build a variety of ω-models with special features, in particular:

Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin.

Corollary 2. For each infinite cardinal κ there are 2κrigid nonisomorphic ω-models of ZFfinof cardinality κ. […]

[BF] S., Baratella and R., Ferro, A theory of sets with the negation of the axiom of infinity, Mathematical Logic Quarterly, vol. 39 (1993), no. 3, pp. 338–352.
[Ba] J., Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1975.
[Be-1] P., Bernays, A system of axiomatic set theory. Part II, The Journal of Symbolic Logic, vol. 6 (1941), pp. 1–17.
[Be-2] P., Bernays, A system of axiomatic set theory. Part VII, The Journal of Symbolic Logic, vol. 19 (1954), pp. 81–96.
[CK] C. C., Chang and H. J., Keisler, Model Theory, North-Holland, Amsterdam, 1973.
[Fr] R., Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Mathematica, vol. 6 (1939), pp. 239–250.
[HP] P., Hájek and P., Pudlák, Metamathematics of First-Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.
[HV] P., Hájek and P., Vopěnka, Über die Gültigkeit des Fundierungsaxioms in speziellen Systemen der Mengentheorie, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 235–241.
[Ha] K., Hauschild, Bemerkungen, das Fundierungsaxiom betreffend, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 51–56.
[Ho] W., Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
[IT] K, Ikeda and A., Tsuboi, Nonstandard models that are definable in models of Peano arithmetic, Mathematical Logic Quarterly, vol. 53 (2007), no. 1, pp. 27–37.
[Ka] R., Kaye, Tennenbaum's theorem for models of arithmetic, this volume.
[KW] R., Kaye and T., Wong, On interpretations of arithmetic and set theory, Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 4, pp. 497–510.
[KS] R., Kossak and J. H., Schmerl, The Structure of Models of Peano Arithmetic, Oxford Logic Guides, vol. 50, The Clarendon Press Oxford University Press, Oxford, 2006, Oxford Science Publications.
[Kr] G., Kreisel, Note on arithmetic models for consistent formulae of the predicate calculus. II, Actes du XIème Congrès International de Philosophie, Bruxelles, 20–26 Août 1953, vol. XIV, North-Holland, Amsterdam, 1953, pp. 39–49.
[Li] P., Lindström, Aspects of Incompleteness, second ed., Lecture Notes in Logic, vol. 10, Association for Symbolic Logic, Urbana, IL, 2003.
[Lo] L., Lovász, Combinatorial Problems and Exercises, AMS Chelsea Publishing, Providence, RI, 2007, Corrected reprint of the 1993 second edition.
[Mac] S. Mac, Lane, Categories for the Working Mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998.
[MZ] A., Mancini and D., Zambella, A note on recursive models of set theories, Notre Dame Journal of Formal Logic, vol. 42 (2001), no. 2, pp. 109–115.
[Moh] S., Mohsenipour, A recursive nonstandard model for open induction with GCD property and cofinal primes, Logic in Tehran, Lecture Notes in Logic, vol. 26, ASL, La Jolla, CA, 2006, pp. 227–238.
[Mos] A., Mostowski, On a system of axioms which has no recursively enumerable arithmetic model, Fundamenta Mathematicae, vol. 40 (1953), pp. 56–61.
[NP] J., Nešetřil and A., Pultr, A note on homomorphism-independent families, Combinatorics (Prague, 1998), Discrete Mathematics, vol. 235 (2001), no. 1-3, pp. 327–334.
[P] E. A., Perminov, The number of rigid graphs that are mutually nonimbeddable into one another, Izvestiya Vysshikh Uchebnykh Zavedeniî. Matematika, (1985), no. 8, pp. 77–78, 86.
[Ra] M., Rabin, On recursively enumerable and arithmetic models of set theory, The Journal of Symbolic Logic, vol. 23 (1958), pp. 408–416.
[Ri] L., Rieger, A contribution to Gödel's axiomatic set theory, Czechoslovak Mathematical Journal, vol. 7 (1957), pp. 323–357.
[Sch-1] J. H., Schmerl, An axiomatization for a class of two-cardinal models, The Journal of Symbolic Logic, vol. 42 (1977), no. 2, pp. 174–178.
[Sch-2] J. H., Schmerl, Tennenbaum's theorem and recursive reducts, this volume.
[Sco] D., Scott, On a theorem of Rabin, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math., vol. 22 (1960), pp. 481–484.
[Vi-1] A., Visser, Faith & Falsity, Annals of Pure and Applied Logic, vol. 131 (2005), no. 1-3, pp. 103–131.
[Vi-2] A., Visser, Categories of theories and interpretations, Logic in Tehran, Lecture Notes in Logic, vol. 26, ASL, La Jolla, CA, 2006, pp. 284–341.
[Vo-1] P., Vopěnka, Axiome der Theorie endlicher Mengen, Československá Akademie Věd. Časopis Pro Pěstování Matematiky, vol. 89 (1964), pp. 312–317.
[Vo-2] P., Vopěnka, Mathematics in the Alternative Set Theory, Teubner-Verlag, Leipzig, 1979.