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Definability of arithmetical operations from binary quadratic forms. (English) Zbl 1024.03011

The author shows that some binary quadratic forms \(F_{a,b,c}(x,y)=ax^2+bxy+cy^2\) are def-complete, i.e. they suffice to define addition and multiplication on the set \(\mathbb N\) of nonnegative integers.

MSC:

03B25 Decidability of theories and sets of sentences
11C99 Polynomials and matrices
03B10 Classical first-order logic
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References:

[1] P. Cegielski: Definability, decidability and complexity. Annals of Mathematics on Artificial Intelligence 16 (1996), 311-341. · Zbl 0865.03007
[2] S. Grigorieff: Decidability et complexite des theories logiques. Logique et Informatique: Une Introduction (B. Courcelle - M. Nivat, I.N.R.I.A., Rocquancourt - France, 1991, pp. 7-97.
[3] I. Korec: List of structures strongest with respect to the first order definability. Preprint 33/1996 of Math. Institute SAV Bratislava, 32pp, latest revision: November 1997, 34pp.
[4] I. Korec: Definability of addition from multiplication and neighborhood relation and some related results. Proceedings of the Conference on Analytic and Elementary Number Theory, Vienna, July 18-20, 1996 (W. G. Nowak and J. Schoissengeier, Universitat fur Bodenkultur and Universitat Wien, 1996, pp. 137-148, also Preprint 23/1996 of Math. Institute SAV Bratislava. · Zbl 0877.03026
[5] I. Korec: Arithmetical operations strongest with respect to the first order definability. Preprint 12/1997 of Math. Institute SAV Bratislava, 12pp.
[6] J. Robinson: Definability and decision problems in arithmetic. Journal of Symbolic Logic 14 (1949), 98-114. · Zbl 0034.00801
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