By Tammo tom Dieck

ISBN-10: 3540505288

ISBN-13: 9783540505280

This publication is a jewel– it explains vital, worthy and deep issues in Algebraic Topology that you just won`t locate somewhere else, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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Doubt what can be seen here is a manifestation ceaselessly recurring of a deep feeling for the unity of the various "mathematical sciences". If one can take as authentic the traditional maxim "All things are numbers" of the first Pythagoreans, that can be considered as the trace of a first attempt to bring back the geometry and algebra of the times to arithmetic. Although the discovery of the irrationals seemed to close for ever that route, the reaction to which it gave birth in Greek mathematics was a second attempt at synthesis this time taking geometry as the basis, and absorbing among others the solution methods for algebraic equations inherited from the Babylonians.

69 For a more detailed and precise description of finite procedures allowed in metamathematics, one can consult, for instance, the thesis of J. Herbrand [158]. 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.

236-271). This work was hardly noticed at the time; although these results, found again by several authors, have been the object of numerous publications since 1935, their historical importance arises much less from the possibilities for applications, fairly thin no doubt, of this theory, than from the fact that they constituted one of the first examples of careful axiomatic construction. On the other hand, the first results of Cantor on sets which were countable or had the power of the continuum were rapidly to have numerous and important applications right up to the most classical questions of Analysis 52 (without talking naturally of those parts of the Cantorian work that inaugurated General Topology and measure theory; on that see pp.