By J. Hirschfeld, W.H. Wheeler
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Extra resources for Forcing Arithmetic Division Rings
From a more technical point of view, certainly the notion of isomorphic groups for abelian groups is known to Gauss, for groups of permutations to Galois (cf. pp. 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.
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". All these objections were not very systematic; it was down to Brouwer and his school to undertake a complete remould of mathematics guided by similar, but even more radical, principles.
Herbrand . 70 When one is speaking of the consistency of the theory of real numbers, one assumes that this theory is defined axiomatically, without using the theory of sets (or at least abstaining from using certain axioms of this latter, such as the axiom of choice or the axiom of the set of subsets). 1. FOUNDATIONS OF MATHEMATICS; LOGIC; SET THEORY. 41 The proof of the independence of a system of propositions Al , A 2 , ••• , An consists in showing that, for each index i, Ai is not a theorem in the theory T; obtained by taking as axioms the Aj with index j :/: i.