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High School Identities

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In 1969, Polish mathematician and logician, Alfred Tarski asked ifall the identities true in the set of natural numbers involving the constant 1,addition, multiplication, and exponentiation can be derived from the elevenaxioms that are taught at the high school level (High School Identities). In1981 Alex Wilkie negatively solved this problem by constructing an identitythat cannot be proved using these axioms. In this paper we survey resultsconnected with Tarski’s problem.
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Bibliografia
  • Asatryan, G. R.: 2004, A solution to identities problem in 2-element HSI-algebras, Math. Log. Quart. 50, 175-178.
  • Baldwin, J. T.: 2010, Logic across the high school curriculum, preprint, University of Illinois at Chicago. http://homepages.math.uic.edu/~jbaldwin/pub/logicaug20.
  • pdf [dostep: 2016-01-29].
  • Birkhoff, G.: 1942, Generalized arithmetic, Duke Math. J. 9, 283-302.
  • Burris, S., Lee, S.: 1993, Tarski’s High School Identities, Amer. Math. Monthly 100, 231-236.
  • Burris, S., Yeats, K. A.: 2004, The saga of the High School Identities, Algebra Universalis 52, 325-342.
  • Dedekind, R.: 1888, Was sind und was sollen die Zahlen?, Verlag Friedrich Vieweg & Sohn, Braunschweig. [tłum. ang.: 1995, What are numbers and what should they be?
  • Revised, edited, and translated from the German by H. Pogorzelski, W. Ryan and W. Snyder. RIM Monographs in Mathematics. Research Institute for Mathematics, Orono, ME].
  • Doner, J., Tarski, A.: 1969, An extended arithmetic of ordinal numbers, Fund. Math. 65, 95-127.
  • Dyer-Bennet, J.: 1940, A theorem on partitions of the set of positive integers, Amer. Math. Monthly 47, 152-154.
  • Gurevic, R.: 1985, Equational theory of positive numbers with exponentiation, Proc. Amer. Math. Soc. 94, 135-141.
  • Gurevic, R.: 1990, Equational theory of positive numbers with exponentiation is not finitely axiomatizable, Ann. Pure Appl. Logic 49, 1-30.
  • Hardy, G. H.: 1910, Orders of Infinity. The ‘Infinitärcalcül’ of Paul Du Bois-Reymond, Cambridge University Press, Cambridge.
  • Henkin, L.: 1977, The logic of equality, Amer. Math. Monthly 84, 597-612.
  • Macintyre, A.: 1981, The laws of exponentiation, w: C. Berline, K. McAloon, J.-P. Ressayre (red.), Model Theory and Arithmetic, Lecture Notes in Math. 890, Springer,
  • Berlin, 185-197.
  • Martin, C. F.: 1973, Equational Theories of Natural Numbers and Transfinite Ordinals, Ph.D. Thesis, University of California, Berkeley, CA.
  • Wilkie, A. J.: 1981, On exponentiation - a solution to Tarski’s high school algebra problem, preprint, Oxford University, 2000, w: A. Macintyre (red.), Connections between Model Theory and Algebraic and Analytic Geometry, Quaderni di Matematica 6, Naples, 107-129. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.
  • 1.15.9695&rep=rep1&type=pdf [dostep: 2016-01-29].
  • Zhang, J.: 2005, Computer search for counterexamples to Wilkie’s Identity, w: R. Nieuwenhuis (red.), Automated Deduction – CADE-20, Lecture Notes in Computer Science 3632, Springer, Berlin, 441-451.
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bwmeta1.element.ojs-issn-2450-341X-year-2015-volume-7-article-3633
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