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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

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