Zermelo s Axiom of Choice

"This book chronicles the work of mathematician Ernst Zermelo (1871-1953) and his development of set theory's crucial principle, the axiom of choice.

Author: Gregory H. Moore

Publisher: Courier Corporation

ISBN: 9780486488417

Category: Mathematics

Page: 410

View: 318

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"This book chronicles the work of mathematician Ernst Zermelo (1871-1953) and his development of set theory's crucial principle, the axiom of choice. It covers the axiom's formulation during the early 20th century, the controversy it engendered, and its current central place in set theory and mathematical logic. 1982 edition"--

Zermelo s Axiom of Choice

When Zermelo's appeared, he thought it less informative than his own. Through correspondence with Russell, he gradually understood how deeply the Axiom of Choice was embedded in his own thinking, and so he accepted the Axiom as valid.

Author: G.H. Moore

Publisher: Springer Science & Business Media

ISBN: 9781461394785

Category: Mathematics

Page: 412

View: 616

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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.

Ernst Zermelo

74 He shows a particular interest in the independence of the axiom of choice (ibid, 3) and gives good advice on this point. When Fraenkel published his results, he thanked Zermelo for providing helpful arguments (1922a, fn. 3).

Author: Heinz Dieter Ebbinghaus

Publisher: Springer

ISBN: 9783662479971

Category: Mathematics

Page: 384

View: 302

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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.

Cantorian Set Theory and Limitation of Size

Zermelo , again in reply to Peano , notes that the axiom of choice for finite sets ( of non - empty sets ) follows from one of Peano's logical axioms ; thus his ( Zermelo's ) axiom is an extension of Peano's axiom to the infinite case .

Author: Michael Hallett

Publisher: Oxford University Press

ISBN: 0198532830

Category: Mathematics

Page: 343

View: 685

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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.

Encyclopaedia of Mathematics

AMS 1980 Subject Classification: 03A05 ZERMELO AXIOM - The axiom of choice for an arbitrary (not necessarily disjoint) family of sets. E. Zermelo stated this axiom in 1904 in the form of the following assertion, which he called the ...

Author: Michiel Hazewinkel

Publisher: Springer Science & Business Media

ISBN: 9789401512336

Category: Mathematics

Page: 536

View: 801

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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.