## Journal of Symbolic Logic

### Theories of arithmetics in finite models

#### Abstract

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1—theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation.

We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication.

#### Article information

Source
J. Symbolic Logic Volume 70, Issue 1 (2005), 1-28.

Dates
First available in Project Euclid: 1 February 2005

http://projecteuclid.org/euclid.jsl/1107298508

Digital Object Identifier
doi:10.2178/jsl/1107298508

Mathematical Reviews number (MathSciNet)
MR2119121

Zentralblatt MATH identifier
05004786

#### Citation

Krynicki, Michał; Zdanowski, Konrad. Theories of arithmetics in finite models. Journal of Symbolic Logic 70 (2005), no. 1, 1--28. doi:10.2178/jsl/1107298508. http://projecteuclid.org/euclid.jsl/1107298508.

#### References

• A. Atserias and Ph. Kolaitis First order logic vs. fixed point logic on finite set theory, 14th IEEE Symposium on Logic in Computer Science (LICS), vol. 14,1999, pp. 275--284.
• D. A. Mix Barrington, N. Immerman, and H. Straubing On uniformity within NC$^1$, Journal of Computer and System Science, vol. 41 (1990), pp. 274--306.
• J. H. Bennett On spectra, Ph.D. thesis, Princeton University,1962.
• W. Bés On Pascal triangles modulo a prime power, Annals of Pure and Applied Logic, vol. 89 (1997), pp. 17--35.
• J. R. Büchi Weak second-order arithmetic and finite automata, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 6 (1960), pp. 66--92.
• M. Davis Hilbert's tenth problem is unsolvable, American Mathematical Monthly, vol. 80 (1973), pp. 233--269.
• A. Dawar, K. Doets, S. Lindell, and S. Weinstein Elementary properties of the finite ranks, Mathematical Journal Quarterly, vol. 44 (1998), pp. 349--353.
• H.-D. Ebbinghaus, J. Flum, and W. Thomas Mathematical logic, second ed., Springer-Verlag,1994.
• P. Hájek and P. Pudlák Metamathematics of first-order arithmetic, Springer-Verlag,1993.
• L. A. Kołodziejczyk private communication.
• I. Korec Elementary theories of structures containing generalized Pascal triangles modulo a prime, Proceedings of the 5th Conference on Discrete Mathematics and Applications, Blagoevrad (S. Shtrakov and I. Marchev, editors),1995, pp. 91--105.
• T. Lee Arithmetical definability over finite structures, Mathematical Logic Quarterly, vol. 49 (2003), pp. 385--393.
• M. Mostowski On representing concepts in finite models, Mathematical Logic Quarterly, vol. 47 (2001), pp. 513--523.
• M. Mostowski and A. Wasilewska Elementary properties of divisibility in finite models, Mathematical Logic Quarterly, vol. 50 (2004), pp. 169--174.
• M. Mostowski and K. Zdanowski FM-representability and beyond, in preparation.
• J. Mycielski Analysis without actual infinity, Journal of Symbolic Logic, vol. 46 (1981), pp. 625--633.
• M. B. Nathanson Elementary methods in number theory, Springer,2000.
• W. Quine Concatenation as a basis for arithmetic, Journal of Symbolic Logic, vol. 11 (1946), pp. 105--114.
• N. Schweikardt On the expressive power of first-order logic with built-in predicates, Ph.D. thesis, Johannes Gutenberg-Universität, Mainz,2001.
• A. L. Semenov Logical theories of one-place functions on the set of natural numbers, Izv. Akad. Nauk SSSR ser. Mat., vol. 47 (1983), pp. 623--658.
• J. R. Shoenfield Recursion theory, Lecture Notes in Logic, Springer-Verlag,1993.
• C. Smoryński Logical number theory I, Springer,1981.
• K. Zdanowski Arithmetic in finite but potentially infinite worlds, Ph.D. thesis, Warsaw University,2004.