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Tarski and geometry

Published online by Cambridge University Press:  12 March 2014

L. W. Szczerba
L. W. Szczerba*
Affiliation:
Institute of Mathematics, Warsaw University, Warsaw, Poland
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Extract

Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.

Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 4 , December 1986 , pp. 907 - 912
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

Gupta, H. N. [1965] Contributions to the axiomatic foundations of geometry, Ph.D. Thesis, University of California, Berkeley, California.Google Scholar
Hilbert, D. [1922] Grundlagen der Geometrie, 3rd ed., Teubner, Leipzig.CrossRefGoogle Scholar
Huntington, E. V. [1916] A set of postulates for abstract geometry exposed in terms of the simple relation of inclusion, Mathematische Annalen, vol. 73, pp. 522–559.CrossRefGoogle Scholar
Jaśkowski, S. [1948] Une modification des définitions foundamentales de la géométrie des corps de M. A. Tarski, Annales de la Société Polonaise de Mathématique, vol. 21, pp. 298–301.Google Scholar
Luschei, E. C. [1962] The logical systems of Leśniewski, North-Holland, Amsterdam.Google Scholar
Makowiecka, H. [1977] On minimal systems of primitives in elementary Euclidean geometry, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 13, pp. 269–277.Google Scholar
Pieri, M. [1908] La geometria elementare instituita sulle nozione di “punto” e “sfera”, Memorie di Matematica e di Fisica della Società Italiana delle Scienze, ser. 3, vol. 15, pp. 345–450.Google Scholar
Royden, H. L. [1959] Remarks on primitive notions for elementary Euclidean and non-Euclidean plane geometry, The axiomatic method (proceedings of the 1957/58 international symposium, Berkeley, California; Henkin, L.et al., editors), North-Holland, Amsterdam, pp. 86–96.Google Scholar
Scott, D. [1956] A symmetric primitive notion for Euclidean geometry, Indagationes Mathematicae, vol. 18, pp. 456–461.CrossRefGoogle Scholar
Sierpiński, W. [1954] On the congruence of sets and their equivalence by finite decomposition, Lucknow University Studies, no. 20, Lucknow University, Lucknow, 1954; reprinted in Congruence of sets and other monographs, Chelsea, New York, 1967.Google Scholar
Szczerba, L. W. [1970] Independence of Pasch's axiom, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 18, pp. 491–498.Google Scholar
Szczerba, L. W. and Szmielew, W. [1970] On the Euclidean geometry without the Pasch axiom, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 18, pp. 659–666.Google Scholar
Szmielew, W. [1959] Some metamathematical problems concerning elementary hyperbolic geometry, The axiomatic method (proceedings of the 1957/58 international symposium, Berkeley, California; Henkin, L.et al., editors), North-Holland, Amsterdam, pp. 30–32.Google Scholar
Szmielew, W. [1974] The role of the Pasch axiom in the foundations of Euclidean geometry, Proceedings of the Tarski symposium (Henkin, L.et al., editors), Proceedings of Symposia in Pure Mathematics, vol. 25, American Mathematical Society, Providence, Rhode Island, pp. 123–132.CrossRefGoogle Scholar
Szmielew, W. [1983] From affine to Euclidean geometry, Pańistwowe Wydawnictwo Naukowe, Warsaw, and Reidel, Dordrecht.Google Scholar