## Axiomatic (and non-axiomatic) mathematics.(English)Zbl 1525.03055

This nicely written article surveys what is known about the problem of axiomatizability for the natural numbers, the integers, the rational numbers, the real numbers, and the complex numbers, seen as structures over languages that contain some combination of addition, multiplication, linear order and exponential. It also discusses some results about the axiomatizability of identities, namely, theories made of universal sentences. An effort is made so that it can be read by non-logicians, though I believe that some basic knowledge of first-order logic is required too fully understand the paper.

### MSC:

 03B25 Decidability of theories and sets of sentences 03B30 Foundations of classical theories (including reverse mathematics) 03C10 Quantifier elimination, model completeness, and related topics 03F40 Gödel numberings and issues of incompleteness 11U05 Decidability (number-theoretic aspects) 12L05 Decidability and field theory

### Keywords:

axiomatic system; first-order logic; identities
Full Text:

### References:

 [1] Z. Assadi and S. Salehi, “On decidability and axiomatizability of some ordered structures”, Soft Comput. 23:11 (2019), 3615-3626. · Zbl 1418.03171 · doi:10.1007/s00500-018-3247-1 [2] J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, Ergebnisse der Math. (3) 36, Springer-Verlag, Berlin, 1998. · Zbl 0912.14023 · doi:10.1007/978-3-662-03718-8 [3] P. Cegielski, “Théorie élémentaire de la multiplication des entiers naturels”, pp. 44-89 in Model theory and arithmetic (Paris, 1979-1980), Lecture Notes in Math. 890, Springer, 1981. · Zbl 0478.03011 · doi:10.1007/BFb0095657 [4] H. B. Enderton, A mathematical introduction to logic, 2nd ed., Academic Press, 2001. · Zbl 0992.03001 [5] R. Gurevič, “Equational theory of positive numbers with exponentiation”, Proc. Amer. Math. Soc. 94:1 (1985), 135-141. · Zbl 0572.03014 · doi:10.2307/2044966 [6] R. Gurevič, “Equational theory of positive numbers with exponentiation is not finitely axiomatizable”, Ann. Pure Appl. Logic 49:1 (1990), 1-30. · Zbl 0707.03023 · doi:10.1016/0168-0072(90)90049-8 [7] R. H. Gurevič, “Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)”, Trans. Amer. Math. Soc. 336:1 (1993), 1-67. · Zbl 0778.30005 · doi:10.2307/2154337 [8] L. Henkin, “The logic of equality”, Amer. Math. Monthly 84:8 (1977), 597-612. · Zbl 0376.02017 · doi:10.2307/2321009 [9] C. W. Henson and L. A. Rubel, “Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions”, Trans. Amer. Math. Soc. 282:1 (1984), 1-32. Corrigendum at 294:1 (1986), 381. · Zbl 0533.03015 · doi:10.2307/1999575 [10] G. Kreisel and J.-L. Krivine, Elements of mathematical logic. Model theory, North-Holland, Amsterdam, 1967. · Zbl 0155.33801 [11] A. Macintyre and A. J. Wilkie, “On the decidability of the real exponential field”, pp. 441-467 in Kreiseliana: about and around Georg Kreisel, edited by P. Odifreddi, A K Peters, Wellesley, MA, 1996. · Zbl 0896.03012 [12] D. Marker, “Model theory and exponentiation”, Notices Amer. Math. Soc. 43:7 (1996), 753-759. · Zbl 1045.03519 [13] C. F. Martin, “Axiomatic bases for equational theories of natural numbers (abstract)”, Notices Amer. Math. Soc. 19:7 (1972), 778-779. [14] B. Poonen, “Undecidability in number theory”, Notices Amer. Math. Soc. 55:3 (2008), 344-350. · Zbl 1194.03018 [15] J. Robinson, “Definability and decision problems in arithmetic”, J. Symbolic Logic 14 (1949), 98-114. · Zbl 0034.00801 · doi:10.2307/2266510 [16] S. Salehi, “On axiomatizability of the multiplicative theory of numbers”, Fund. Inform. 159:3 (2018), 279-296. · Zbl 1436.03307 · doi:10.3233/fi-2018-1665 [17] S. Salehi and M. Zarza, “First-order continuous induction and a logical study of real closed fields”, Bull. Iranian Math. Soc. 46:1 (2020), 225-243. · Zbl 1486.03060 · doi:10.1007/s41980-019-00252-0 [18] K. Schmüdgen, “Around Hilbert’s 17th problem”, Doc. Math. Extra volume: Optimization stories (2012), 433-438. · Zbl 1268.14052 [19] C. Smoryński, Logical number theory, I: An introduction, Springer, Berlin, 1991. · Zbl 0759.03002 · doi:10.1007/978-3-642-75462-3 [20] A. J. Wilkie, “On exponentiation: a solution to Tarski’s high school algebra problem”, preprint. Reprinted in Connections between model theory and algebraic and analytic geometry, Quaderni di Matematica (Seconda Univ. Napoli), 6, (2000), pp. 107-129 · Zbl 0993.03044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.