High School Identities

Katarzyna Słomczyńska

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia (2015)

  • Volume: 7, page 91-98
  • ISSN: 2080-9751

Abstract

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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.

How to cite

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Katarzyna Słomczyńska. "High School Identities." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 7 (2015): 91-98. <http://eudml.org/doc/296270>.

@article{KatarzynaSłomczyńska2015,
abstract = {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.},
author = {Katarzyna Słomczyńska},
journal = {Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia},
keywords = {HSI-algebras; exponentiation; Wilkie’s identity},
language = {pol},
pages = {91-98},
title = {High School Identities},
url = {http://eudml.org/doc/296270},
volume = {7},
year = {2015},
}

TY - JOUR
AU - Katarzyna Słomczyńska
TI - High School Identities
JO - Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
PY - 2015
VL - 7
SP - 91
EP - 98
AB - 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.
LA - pol
KW - HSI-algebras; exponentiation; Wilkie’s identity
UR - http://eudml.org/doc/296270
ER -

References

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  3. pdf [dostep: 2016-01-29]. 
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  12. Gurevic, R.: 1990, Equational theory of positive numbers with exponentiation is not finitely axiomatizable, Ann. Pure Appl. Logic 49, 1-30. 
  13. Hardy, G. H.: 1910, Orders of Infinity. The ‘Infinitärcalcül’ of Paul Du Bois-Reymond, Cambridge University Press, Cambridge. 
  14. Henkin, L.: 1977, The logic of equality, Amer. Math. Monthly 84, 597-612. 
  15. 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, 
  16. Berlin, 185-197. 
  17. Martin, C. F.: 1973, Equational Theories of Natural Numbers and Transfinite Ordinals, Ph.D. Thesis, University of California, Berkeley, CA. 
  18. 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. 
  19. 1.15.9695&rep=rep1&type=pdf [dostep: 2016-01-29]. 
  20. 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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