## Notre Dame Journal of Formal Logic

### On Interpretations of Bounded Arithmetic and Bounded Set Theory

Richard Pettigrew

#### Abstract

In "On interpretations of arithmetic and set theory," Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

Theorem The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic $I Δ0 +exp$. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of the arithmetic in the set theory. Instead, I am forced to produce a different interpretation.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 50, Number 2 (2009), 141-151.

Dates
First available in Project Euclid: 11 May 2009

https://projecteuclid.org/euclid.ndjfl/1242067707

Digital Object Identifier
doi:10.1215/00294527-2009-003

Mathematical Reviews number (MathSciNet)
MR2535581

Zentralblatt MATH identifier
1183.03029

Subjects

#### Citation

Pettigrew, Richard. On Interpretations of Bounded Arithmetic and Bounded Set Theory. Notre Dame J. Formal Logic 50 (2009), no. 2, 141--151. doi:10.1215/00294527-2009-003. https://projecteuclid.org/euclid.ndjfl/1242067707

#### References

• [1] Ackermann, W., "Die Widerspruchsfreiheit der allgemeinen Mengenlehre", Mathematische Annalen, vol. 114 (1937), pp. 305--15.
• [2] 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 Monograph. Enseign. Math., Univerisité de Genève, Geneva, 1982.
• [3] Homolka, V., A System of Finite Set Theory Equivalent to Elementary Arithmetic, Ph.D. thesis, University of Bristol, 1983.
• [4] 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.
• [5] Mayberry, J. P., The Foundations of Mathematics in the Theory of Sets, vol. 82 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2000.
• [6] Mayberry, J. P., and R. Pettigrew, "Natural number systems in the theory of finite sets", http://arxiv.org/abs/0711.2922, 2007.
• [7] Parikh, R., "Existence and feasibility in arithmetic", The Journal of Symbolic Logic, vol. 36 (1971), pp. 494--508.