Zermelo’s Axiom of Choice

Zermelo’s Axiom of Choice PDF Author: G.H. Moore
Publisher: Springer Science & Business Media
ISBN: 1461394783
Category : Mathematics
Languages : en
Pages : 425

Get Book

Book Description
This book grew out of my interest in what is common to three disciplines: mathematics, philosophy, and history. The origins of Zermelo's Axiom of Choice, as well as the controversy that it engendered, certainly lie in that intersection. Since the time of Aristotle, mathematics has been concerned alternately with its assumptions and with the objects, such as number and space, about which those assumptions were made. In the historical context of Zermelo's Axiom, I have explored both the vagaries and the fertility of this alternating concern. Though Zermelo's research has provided the focus for this book, much of it is devoted to the problems from which his work originated and to the later developments which, directly or indirectly, he inspired. A few remarks about format are in order. In this book a publication is indicated by a date after a name; so Hilbert 1926, 178 refers to page 178 of an article written by Hilbert, published in 1926, and listed in the bibliography.

Zermelo’s Axiom of Choice

Zermelo’s Axiom of Choice PDF Author: G.H. Moore
Publisher: Springer Science & Business Media
ISBN: 1461394783
Category : Mathematics
Languages : en
Pages : 425

Get Book

Book Description
This book grew out of my interest in what is common to three disciplines: mathematics, philosophy, and history. The origins of Zermelo's Axiom of Choice, as well as the controversy that it engendered, certainly lie in that intersection. Since the time of Aristotle, mathematics has been concerned alternately with its assumptions and with the objects, such as number and space, about which those assumptions were made. In the historical context of Zermelo's Axiom, I have explored both the vagaries and the fertility of this alternating concern. Though Zermelo's research has provided the focus for this book, much of it is devoted to the problems from which his work originated and to the later developments which, directly or indirectly, he inspired. A few remarks about format are in order. In this book a publication is indicated by a date after a name; so Hilbert 1926, 178 refers to page 178 of an article written by Hilbert, published in 1926, and listed in the bibliography.

Zermelo's Axiom of Choice

Zermelo's Axiom of Choice PDF Author: Gregory H. Moore
Publisher:
ISBN: 9783540906704
Category : Auswahlaxiom
Languages : en
Pages : 410

Get Book

Book Description


The Axiom of Choice

The Axiom of Choice PDF Author: John Lane Bell
Publisher: Studies in Logic. Mathematical
ISBN: 9781904987543
Category : Mathematics
Languages : en
Pages : 248

Get Book

Book Description
This book presents an overview of the development of the Axiom of Choice since its introduction by Zermelo at the beginning of the last century. The book surveys the Axiom of Choice from three perspectives. The first, or mathematical perspective, is that of the "working mathematician". This perspective brings into view the manifold applications of the Axiom of Choice-usually in the guise of Zorn s Lemma- in a great variety of areas of mathematics. The second, foundational, perspective is that of the logician or constructive mathematician concerned with the foundational status of the Axiom of Choice. The third, topos-theoretical, perspective is that taken by the mathematician or logician investigating the role of the Axiom of Choice in topos theory. Certain topics-for instance mathematical applications of the Axiom, and its relationship with logic-are discussed in considerable detail. Others-notably the consistency and independence of the Axiom of the usual systems of set theory-are given no more than summary treatment, the justification here being that these topics have been given full expositions elsewhere. It is hoped that the book will be of interest to logicians and mathematicians, both professional and prospective.

Zermelo's Axiom of Choice

Zermelo's Axiom of Choice PDF Author: G. H. Moore
Publisher:
ISBN: 9781461394792
Category :
Languages : en
Pages : 428

Get Book

Book Description


Mathematics

Mathematics PDF Author: Morris Kline
Publisher: Oxford University Press, USA
ISBN: 9780195030853
Category : Mathematics
Languages : en
Pages : 380

Get Book

Book Description
This work stresses the illogical manner in which mathematics has developed, the question of applied mathematics as against 'pure' mathematics, and the challenges to the consistency of mathematics' logical structure that have occurred in the twentieth century.

Cantorian Set Theory and Limitation of Size

Cantorian Set Theory and Limitation of Size PDF Author: Michael Hallett
Publisher: Oxford University Press
ISBN: 9780198532835
Category : Mathematics
Languages : en
Pages : 372

Get Book

Book Description
Cantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Frankel set theory.

From Frege to Gödel

From Frege to Gödel PDF Author: Jean van Heijenoort
Publisher: Harvard University Press
ISBN: 0674257243
Category : Philosophy
Languages : en
Pages : 684

Get Book

Book Description
The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded by Leibniz, may be said to have been initiated by Boole, De Morgan, and Jevons, but it was the publication in 1879 of Gottlob Frege’s Begriffsschrift that opened a great epoch in the history of logic by presenting, in full-fledged form, the propositional calculus and quantification theory. Frege’s book, translated in its entirety, begins the present volume. The emergence of two new fields, set theory and foundations of mathematics, on the borders of logic, mathematics, and philosophy, is depicted by the texts that follow. Peano and Dedekind illustrate the trend that led to Principia Mathematica. Burali-Forti, Cantor, Russell, Richard, and König mark the appearance of the modern paradoxes. Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory. Skolem generalizes Löwenheim’s theorem, and he and Fraenkel amend Zermelo’s axiomatization of set theory, while von Neumann offers a somewhat different system. The controversy between Hubert and Brouwer during the twenties is presented in papers of theirs and in others by Weyl, Bernays, Ackermann, and Kolmogorov. The volume concludes with papers by Herbrand and by Gödel, including the latter’s famous incompleteness paper. Of the forty-five contributions here collected all but five are presented in extenso. Those not originally written in English have been translated with exemplary care and exactness; the translators are themselves mathematical logicians as well as skilled interpreters of sometimes obscure texts. Each paper is introduced by a note that sets it in perspective, explains its importance, and points out difficulties in interpretation. Editorial comments and footnotes are interpolated where needed, and an extensive bibliography is included.

Zermelo's Well-ordering Theorem and the Axiom of Choice

Zermelo's Well-ordering Theorem and the Axiom of Choice PDF Author: Craig Nicholas Bach
Publisher:
ISBN:
Category :
Languages : en
Pages : 70

Get Book

Book Description


Encyclopaedia of Mathematics

Encyclopaedia of Mathematics PDF Author: Michiel Hazewinkel
Publisher: Springer Science & Business Media
ISBN: 1556080085
Category : Mathematics
Languages : en
Pages : 556

Get Book

Book Description
This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fme subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.

Ernst Zermelo

Ernst Zermelo PDF Author: Heinz Dieter Ebbinghaus
Publisher: Springer
ISBN: 3662479974
Category : Mathematics
Languages : en
Pages : 384

Get Book

Book Description
This biography sheds light on all facets of the life and the achievements of Ernst Zermelo (1871-1953). Zermelo is best-known for the statement of the axiom of choice and his axiomatization of set theory. However, he also worked in applied mathematics and mathematical physics. His dissertation, for example, promoted the calculus of variations, and he created the pivotal method in the theory of rating systems. The presentation of Zermelo's work explores motivations, aims, acceptance, and influence. Selected proofs and information gleaned from letters add to the analysis. The description of his personality owes much to conversations with his late wife Gertrud. This second edition provides additional information. The system of citations has been adapted to that of Zermelo's Collected Works in order to facilitate side-by-side reading and thus profit from the thorough commentaries written for the Collected Works by experts in the respective fields. All facts presented are documented by appropriate sources. The biography contains nearly 50 photos and facsimiles.