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)

Bibliography of A. A. Fraenkel (p. ix-x)--Axiomatic set theory. Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, von P. Bernays.--On some problems involving inaccessible cardinals, by P. Erdös and A. Tarski.--Comparing the axioms of local and universal choice, by A. Lévy.--Frankel's addition to the axioms of Zermelo, by R. Mantague.--More on the axiom of extensionality, by D. Scott.--The problem of predicativity, by J. R. Shoenfield.--Mathematical logic. Grundgedanken einer typenfreien Logik, von W. Ackermann.--On the use of Hilbert's [epsilon]-operator in scientific (...) theories, by R. Carnap.--Basic verifiability in the combinatory theory of restricted generality, by H. B. Curry.--Uniqueness ordinals in constructive number classes, by H. Putnam.--On the construction of models, by A. Robinson.--Interpretation of mathematical theories in the first order predicate calculus, by T. Skolem.--The elementary character of two notions from general algebra, by R. Vaught.--Foundations of arithmetic and analysis. Axiomatic method and intuitionism, by A. Heyting.--On rank-decreasing functions, by G. Kurepa.--On non-standard models for number theory, by E. Mendelson.--Concerning the problem of axiomatizability of the field of real numbers in the weak second order logic, by A. Mostowski.--Non-standard models and independence of the induction axiom, by M. O. Rabin.--Sur les ensembles raréfiés de nombres naturels, par W. Sierpinski.--Philosophy of logic and mathematics. Remarks on the paradoxes of logic and set theory, by E. W. Beth.--Logique formalisée et raisonnement juridique, par R. Feys.--Im Umkreis der sogenannten Raumprobleme, von H. Freudenthal.--Process and existence in mathematics, by H. Wang. (shrink)

In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-calledGrundlagenstreitwas in full swing. An emotional man like (...) Brouwer, who easily suffered under stress, was already on edgewhen Fraenkel's bookZehn Vorlesungen Über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear.With theGrundlagenstreitreaching (in print!) a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer's reactions. since the Fraenkel letters have not been preserved.The late Mrs. Fraenkel kindly put the Brouwer letters that were in her possession at my disposal. I am grateful to the Fraenkel family for the permission to use the material.I am indebted to Andreas Blass for his valuable suggestions and corrections.Abraham Fraenkel (then still called Adolf) was one of the first non-partisan mathematicians, if not the first, who developed a genuine interest in intuitionism. (shrink)

Officially inaugurated in 1925, the Hebrew University of Jerusalem was designed to serve the academic needs of the Jewish people and the Zionist enterprise in British Mandatory Palestine, as well as to help fulfill the economic and social requirements of the Middle East. It is intriguing that a university with such practical goals should have as one of its central pillars an institute for pure mathematics that purposely dismissed any of the varied fields of applied mathematics. This paper tells of (...) the preparations for the inauguration of the Hebrew University during the years 1920–1925 and analyzes the founding phase of the Einstein Institute of Mathematics that was established there during the years 1924–1928. Special emphasis is given to the first terms in which this Institute operated, starting from the winter of 1927 with the activities of the director and one of the founders, the German mathematician Edmund Landau, and onward from 1928 when his successors, particularly Adolf Abraham Halevi Fraenkel and Mihály-Michael Fekete, continued Landau's heritage of pure mathematics. The paper shows why and how the Institute succeeded in rejecting applied mathematics from its court and also explores the controversial issue of center and periphery in the development of science, a topic that is briefly analyzed in the concluding section. (shrink)

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit was in full swing. An emotional (...) man like Brouwer, who easily suffered under stress, was already on edgewhen Fraenkel's book Zehn Vorlesungen Über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear.With the Grundlagenstreit reaching a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer's reactions. since the Fraenkel letters have not been preserved.The late Mrs. Fraenkel kindly put the Brouwer letters that were in her possession at my disposal. I am grateful to the Fraenkel family for the permission to use the material.I am indebted to Andreas Blass for his valuable suggestions and corrections.Abraham Fraenkel was one of the first non-partisan mathematicians, if not the first, who developed a genuine interest in intuitionism. (shrink)