By S. R. Sario, M. Nakai, C. Wang, L. O. Chung

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35 Following this, with each new axiomatic theory, it is a natural development to define a notion of isomorphism; but it is only with the modern notion of structure that it was finally recognised that every structure carries within itself a notion of isomorphism, and that it is not necessary to give a special definition of it for each type of structure. C) The arithmetisation of classical mathematics. - The use, more and more widespread, of the notion of "model" was also going to allow the realisation in the XIXth century of the unification of mathematics dreamed of by the Pythagoreans.

This latter had been clearly conceived and used for the first time in the XVIIth century by B. Pascal ([244]. v. III. p. 456)37 - even though one can find in Antiquity some more or less conscious applications of it - and was currently used by mathematicians since the second half of the XVIIth century. But it is only in 1888 that Dedekind ([79], v. III. 359-361) stated a complete system of axioms for arithmetic (a system repeated 3 years later by Peano and usually known by his name [246 c)), which contained in As examples of non-commutative operations, he points to subtraction.

66 Can one then speak of the "existence" of a set of which one does not know how to "name" each element? Already Baire does not hesitate to deny the "existence" of the set of all subsets of a given infinite set (loc. , pp. 263-264); in vain does Hadamard observe that these requirements lead to renouncing even the possibility of speaking of the real numbers: it is to just that conclusion that E. Borel finally agrees. Putting aside the fact that countability seems to have acquired freehold, one has returned just about to the classical position of the adversaries of the "actual infinite".