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Interpretations of Euclidean geometry


Author: S. Świerczkowski
Journal: Trans. Amer. Math. Soc. 322 (1990), 315-328
MSC: Primary 03F25; Secondary 03C65, 51M99
DOI: https://doi.org/10.1090/S0002-9947-1990-0982234-9
MathSciNet review: 982234
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Abstract: Following Tarski, we view $ n$-dimensional Euclidean geometry as a first-order theory $ {E_n}$ with an infinite set of axioms about the relations of betweenness (among points on a line) and equidistance (among pairs of points). We show that for $ k < n$, $ {E_n}$ does not admit a $ k$-dimensional interpretation in the theory RCF of real closed fields, and we deduce that $ {E_n}$ cannot be interpreted $ r$-dimensionally in $ {E_s}$, when $ r \cdot s < n$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0982234-9
Keywords: Interpretation, definable set, semialgebraic, Cartesian coordinates
Article copyright: © Copyright 1990 American Mathematical Society

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