## Notre Dame Journal of Formal Logic

### On Interpretations of Arithmetic and Set Theory

#### Abstract

This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the nature of the equivalence of PA and ZF−inf and corrects some errors in the literature. We also survey the restrictions of the Ackermann interpretation and its inverse to subsystems of PA and ZF−inf, where full induction, replacement, or separation is not assumed. The paper concludes with a discussion on the problems one faces when the totality of exponentiation fails, or when the existence of unordered pairs or power sets is not guaranteed.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 48, Number 4 (2007), 497-510.

Dates
First available in Project Euclid: 29 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1193667707

Digital Object Identifier
doi:10.1305/ndjfl/1193667707

Mathematical Reviews number (MathSciNet)
MR2357524

Zentralblatt MATH identifier
1137.03019

#### Citation

Kaye, Richard; Wong, Tin Lok. On Interpretations of Arithmetic and Set Theory. Notre Dame J. Formal Logic 48 (2007), no. 4, 497--510. doi:10.1305/ndjfl/1193667707. https://projecteuclid.org/euclid.ndjfl/1193667707

#### References

• [1] Ackermann, W., "Die Widerspruchsfreiheit der allgemeinen Mengenlehre", Mathematische Annalen, vol. 114 (1937), pp. 305--15.
• [2] Baratella, S., and R. Ferro, "A theory of sets with the negation of the axiom of infinity", Mathematical Logic Quarterly, vol. 39 (1993), pp. 338--52.
• [3] Barwise, J., Admissible Sets and Structures: An Approach to Definability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1975.
• [4] Chang, C. C., and H. J. Keisler, Model Theory, 2d edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1977.
• [5] Diaconescu, R., and L. A. S. Kirby, "Models of arithmetic and categories with finiteness conditions", Annals of Pure and Applied Logic, vol. 35 (1987), pp. 123--48.
• [6] Drake, F. R., Set Theory: An Introduction to Large Cardinals, vol. 76 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1974.
• [7] Friedman, H., "Categoricity with respect to ordinals", pp. 17--20 in Higher Set Theory (Proceedings of the Conference, Mathematisches Forschungsinstitut (Oberwolfach, 1977), edited by G. H. Müller and D. S. Scott, vol. 669 of Lecture Notes in Mathematics, Springer, Berlin, 1978.
• [8] Gaifman, H., and C. Dimitracopoulos, "Fragments of Peano's arithmetic and the MRDP" theorem, pp. 187--206 in Logic and Algorithmic (Zurich, 1980), vol. 30 of Monographies de L'Enseignement Mathématique, Université de Genève, Geneva, 1982.
• [9] Hájek, P., and P. Pudlák, Metamathematics of First-Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1993.
• [10] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, The Clarendon Press, New York, 1991.
• [11] Kunen, K., Set Theory: An Introduction to Independence Proofs, vol. 102 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1980.
• [12] Mancini, A., and D. Zambella, "A note on recursive models of set theories", Notre Dame Journal of Formal Logic, vol. 42 (2001), pp. 109--15.
• [13] Pudlák, P., "Cuts, consistency statements and interpretations", The Journal of Symbolic Logic, vol. 50 (1985), pp. 423--41.
• [14] Ressayre, J.-P., "Modèles non standard et sous-systèmes remarquables de $\ZF$"", pp. 47--147 in Modèles non Standard en Arithmétique et théorie des Ensembles, vol. 22 of Publications Mathématiques de l'Université Paris VII, Université de Paris VII U.E.R. de Mathématiques, Paris, 1987.
• [15] Visser, A., "Categories of theories and interpretations", pp. 284--341 in Logic in Tehran. Proceedings of the Workshop and Conference on Logic, Algebra, and Arithmetic, (October, 2003), edited by A. Enayat, I. Kalantari, and M. Moniri, vol. 26 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2006. Logic Group Preprint Series 228, Faculteit Wijsbegeerte, Universiteit Utrecht, h% ttp://www.phil.uu.nl/preprints/preprints/PREPRINTS/preprint228.pdf.
• [16] Wilkie, A. J., and J. B. Paris, "On the scheme of induction for bounded arithmetic formulas", Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261--302.
• [17] Wilkie, A. J., and J. B. Paris, "On the existence of end extensions of models of bounded induction", pp. 143--61 in Logic, Methodology and Philosophy of Science, VIII (Moscow, 1987), edited by J. E. Fenstad, I. T. Frolov, and R. Hilpinen, vol. 126 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1989.