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Interpretations of Euclidean geometry. (English) Zbl 0715.03004

Let \(E_ n\) be Tarski’s first-order theory with equality for n- dimensional Euclidean geometry, in the language \(L_{\beta \delta}\) with ternary predicate \(\beta\) and quaternary predicate \(\delta\). \(E_ n\) has Euclidean n-dimensional space \({\mathcal E}_ n\) as model, with \(\beta\) (a,b,c) meaning that point b lies between a and c (possibly \(b=a\) or \(b=c)\), and \(\delta\) (a,b,c,d) meaning that the distance between a and b is equal to the distance between c and b. This interpretation has its counterpart in an interpretation of \(E_ n\) in the theory RCF of real- closed fields. (Variables for points are replaced by n-tuples of variables, and the interpretations of \(\beta\) and \(\delta\) can be expressed in terms of the coordinates of points. Theorems of \(E_ n\) translate into theorems of RCF. The language of RCF is the language \(L_{OF}\) of ordered fields.) A. Tarski [Symp. Axiomatic Method, Stud. Logic Found. Math., 16-29 (1959; Zbl 0092.385)] proved that \(E_ n\) is complete. Hence, the theorems of \(E_ n\) are the sentences of \(L_{\beta \delta}\) that are true in \({\mathcal E}_ n\). The author’s main result is:
Theorem 1.1: For every \(n\geq 2\), there exists a theorem \(\phi_ n\) of \(E_ n\) such that, for every \(k<n\), the image of \(\phi_ n\) under any k-dimensional interpretation of the language \(L_{\beta \delta}\) in the language \(L_{OF}\) is not a theorem of RCF.
This confirms a conjecture of J. Mycielski [J. Symb. Logic 42, 297- 305 (1977; Zbl 0371.02026)]. A special case, when \(n=2\) and \(k=1\), was proved by M. Boffa [Bull. Soc. Math. Belg., Sér. B 32, 107-133 (1980; Zbl 0469.03017)]. The author’s proof extends Boffa’s method by using a theorem of S. Lojasiewicz [Ann. Sc. Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 18, 449-474 (1964; Zbl 0128.171)].
Reviewer: E.Mendelson

MSC:

03B30 Foundations of classical theories (including reverse mathematics)
51M05 Euclidean geometries (general) and generalizations
12L99 Connections between field theory and logic
03F25 Relative consistency and interpretations
03C65 Models of other mathematical theories
12J15 Ordered fields
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