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Notation systems and recursive ordered fields

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Notation systems and recursive ordered fields COMPOSITIO MATHEMATICA YIANNISN.MOSCHOVAKIS Notation systems and recursive ordered fields Compositio Mathematica, tome 17 (1965-1966), p. 40-71. <> © Foundation Compositio Mathematica, 1965-1966, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 40 Notation systems and recursive ordered fields 1 by Yiannis N. Moschovakis Introduction The field of real numbers may be introduced in one of two ways. In the so-called "constructive" or "genetic" method [6, p. 26], one defines the real numbers directly from the rational numbers as infinite decimals, Dedekind cuts, Cauchy sequences, nested interval sequences or some other similar objects. In the "axio- matic" or "postulational" method, on the other hand, one simply takes the real numbers to be any system of objects which satisfies the axioms for a "complete ordered field". (If we postulate "Cauchy-completeness" rather than "order-completeness", we must also require the field to be archimedean [4, Ch. II, Sec. s-io].) These two methods do not contradict each other, but are in fact complementary. The Dedekind construction furnishes an existence proof for the axiomatic approach. Similarly, the axio- matic characterization provides a certain justification for the seemingly arbitrary choice of any particular construction; for we can show that any two complete ordered fields are isomorphic [4, Ch. II, Sec. 9-10]. In each of the above-mentioned genetic approach

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