This paper is concerned with the influence that the set theory of Georg Cantor bore upon the mathematical logic of Bertrand Russell. In some respects the influence is positive, and stems directly from Cantor's writings or through intermediary figures such as Peano; but in various ways negative influence is evident, for Russell adopted alternative views about the form and foundations of set theory. After an opening biographical section, six sections compare and contrast their views on matters of common interest; irrational (...) numbers, infinitesimals, cardinal and ordinal numbers, the axiom of infinity, the paradoxes, and the axioms of choice. Two further sections compare the two men over more general questions: the role of logic and the philosophy of mathematics. In a final section I draw some conclusions. (shrink)

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms (...) and proof processes of that formal system. Gentzen's proof succeeds by utilisingtransfinite inductionto prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to provethe restricted ordinal theorem, that a descending sequence of ordinals, less thanε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction isfinitist. (shrink)

One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets.

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)