Abstract
The completeness axiom V for plane Euclidean Geometry leaves us facing the problem already considered in Chapter I: In what sense are the sets M and N named in it to be understood? In other words, how to adequately express the content of Axiom V in our formal language? And once more we choose the same solution - we identify sets with extensions of predicates, in the present case with predicates in the first order language of elementary geometry built the basic concepts introduced in § 2. The resulting completeness axiom, made elementary, is called the Tarski Schema.
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Further Reading
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Tarski, A.: What is Elementary Geometry?, in Henkin, Suppes & Tarski: The Axiomatic Method, pp. 16–29, Amsterdam, North-Holland, 1959
Scott, D.: Dimension in Elementary Euclidean Geometry, in: Henkin, Suppes & Tarski: The Axiomatic Method, pp. 53–67, Amsterdam, North-Holland, 1959
Robinson, R. Binary Relations as Primitive Notions in Elementary Geometry, in: Henkin, Suppes & Tarski: The Axiomatic Method, pp. 68–85, Amsterdam, North- Holland, 1959
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Schwabhäuser, W. & Szczerba, L.W.: Relations on Lines as Primitive Notions for Euclidean Geometry, Fundamenta Mathematicae, vol. 82, pp. 347–355, (1975)
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Engeler, E. (1993). Metatheoretical Questions and Methods in Elementary Geometry. In: Foundations of Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78052-3_8
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DOI: https://doi.org/10.1007/978-3-642-78052-3_8
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