Results 1 
5 of
5
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusi ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
BERNAYS AND SET THEORY
"... We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. ..."
Abstract
 Add to MetaCart
(Show Context)
We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles.
THE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR AKIHIRO KANAMORI
, 2002
"... For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks ..."
Abstract
 Add to MetaCart
(Show Context)
For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of ’ {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary settheoretic concepts serves as a motif that reflects and illuminates larger and more significant developments in mathematical logic:
unknown title
"... In this elementary paper we establish a few novel results in set theory; their interest is wholly foundationalphilosophical in motivation. We show that in CantorVon Neumann SetTheory, which is a reformulation of Von Neumann's original theory of functions and things that does not introduce ..."
Abstract
 Add to MetaCart
In this elementary paper we establish a few novel results in set theory; their interest is wholly foundationalphilosophical in motivation. We show that in CantorVon Neumann SetTheory, which is a reformulation of Von Neumann's original theory of functions and things that does not introduce `classes ' (let alone `proper classes'), developed in the 1920ies, both the Pairing Axiom and `half ' the Axiom of Limitation are redundant the last result is novel. Further we show, in contrast to how things are usually done, that some theorems, notably the Pairing Axiom, can be proved without invoking the Replacement Schema (F) and the PowerSet Axiom. Also the Axiom of Choice is redundant in CVN, because it a theorem of CVN. The philosophical interest of CantorVon Neumann SetTheory, which is very succinctly indicated, lies in the fact that it is far better suited than ZermeloFraenkel SetTheory as an axiomatisation of what Hilbert famously called Cantor's Paradise. From Cantor one needs to jump to Von Neumann, over the heads of Zermelo and Fraenkel, and then reformulate.