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Determinateness and the Pasch Axiom

Published online by Cambridge University Press:  20 November 2018

Andrew Adler
Andrew Adler*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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Let E be the (nonelementary) plane Euclidean geometry without the Pasch axiom. (The Pasch axiom says that a line cutting one side of a triangle must also cut another side. A full list of axoms for E is given in [5].) E satisfies in particular the full second-order continuity axiom.

Szczerba [5] has recently shown using a Hamel basis for the reals over the rationals that there exists a model of E not satisfying the Pasch axiom. It is natural to ask whether the axiom of choice plays an essential role in the proof. It will turn out that it does.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 2 , June 1973 , pp. 159 - 160
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Borsuk, K. and Szmielew, W., Foundations of geometry, North-Holland, Amsterdam, 1960.Google Scholar
2. Hahn, H. and Rosenthal, A., Set functions, Univ. of New Mexico, 1948.Google Scholar
3. Mycielski, J., On the axiom of determinateness, Fund. Math. 53 (1963), 205–224.Google Scholar
4. Mycielski, J. and Swierczkowski, S., On the Lebesgue measurability and the axiom of determinateness, Fund. Math. 54 (1964), 67–71.Google Scholar
5. Szczerba, L.W., Independence of Pasch’s axiom., Bull. Acad. Polon. Sci. Sér. Sci. Tech. (11), 18 (1970), 659–666.Google Scholar
6. Szczerba, L.W. and Szmielew, W., On the Euclidean geometry without the Pasch axiom, Bull. Acad. Pol. Sci. (8) 18 (1970), 491–498.Google Scholar