Abstract
The word ‘theory’ is often understood as dependent on the language used to formalise it (cf. e.g. Henkin et al, 1971, p. 44). As a consequence, the theory of groups formalised without a neutral element symbol is a proper subtheory of that formalised in a language with a symbol for this element. We may observe an even more striking situation in geometry: the systems presented in Pieri (1908) and Tarski (1959) using different primitive notions are completely different theories, though they are intuitively closely related. The series of papers (e.g. Beth and Tarski, 1956; Scott, 1956; Tarski, 1956; R. Robinson, 1959; Royden, 1959) has been devoted to the study of possible systems of primitive notions of Euclidean geometry. Each of these systems may be used to express the same facts in a completely different manner. The difference may even be in the elements of the universes: Tarski (1959) uses points only, Hilbert (1930)-points, lines, planes and angles, Schwabhäuser and Szczerba (1975)-lines only, and Tarski (1929) — open discs or balls. Each of these formalisations may be included in any other by means of proper definitions, the procedure used commonly by most mathematicians. In fact if a lecturer speaking about some mathematical topic needs a new definable notion, no matter whether relation or object, he usually just introduces it, not bothering to change the language; and from that moment he works with a different theory.
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Szczerba, L.W. (1977). Interpretability of Elementary Theories. In: Butts, R.E., Hintikka, J. (eds) Logic, Foundations of Mathematics, and Computability Theory. The University of Western Ontario Series in Philosophy of Science, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1138-9_8
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