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Cantorian Set Theory and Limitation of Size (Oxford Logic Guides)

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This is BY FAR the best and most INTERESTING book available on how Cantor developed his key ideas about transfinite sets, large cardinals, ordinals etc. It contains materials that will be highly relevant to even the most advanced set theorists, while yet managing to be generally accessible to those who, like myself, have only around a B.S. mathematics degree level of understanding of the field. This ability to be of use and interest to readers with such widely varied mathematical preparations is a true tribute to the author's gift for being able to explain even very advanced concepts clearly and directly, something which is evident throughout the text, -- and unfortunately sorely missing in most mathematical texts operating at such a high level of abstraction. To be a bit more precise, I hope, persons with only a basic understanding of set theory -- something around what one should be able to glean from reading, say, Halmos' "Naive Set Theory" -- will indeed find themselves "out at sea" at times, but actually surprisingly FEW times, considering how well the author manages to unpack most of the key concepts and draw you back into the primary narrative. No doubt because this book is so much better than all its competitors, used copies, even the paperbacks, are now selling for a small fortune. [I kid you not, I just saw one listed at over $990!; though a few minutes of searching the main web book seller consolidators -- including the listings here at Amazon -- should still bag you one for under $100, at least if you act reasonably soon.] Clearly the publisher really needs to reissue this work to meet the fully justified demand! When and if they will do so, -- after all, it was last released, in its one and only paperback edition, in 1986! -- is anybody's guess. So, if you are a Cantor scholar, or a serious set theorist of any persuasion, you should probably bite the bullet and buy one now before the price really goes through the roof and you have to rely on marked-up, slowly disappearing library copies, -- until, that is, they all get stolen and resold on the web, which has happened to several classic math works already. One last thing, for those who don't really need the most sophisticated work on Cantor's intellectual development, Joseph Dauben's biography of Cantor is also very good and still widely available at a reasonable price. [His biography of Abraham Robinson is also very good by the way.]

Georg Cantor lived from 1845 to 1918. Judging from Hallet's bibliography, Cantor's publications on sets and infinity occurred in the years 1872 to 1897. A letter of 1899 to Richard Dedekind [1831-1916] is also relevant. This letter is the only writing by Cantor included in Jean van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. No letter or publication by Cantor later than 1899 is discussed in Hallet's book, although a letter from 1903 to Philip Jourdain [1879-1919] is mentioned as outlining the proof given in the 1899 letter to Dedekind, and a partial sentence is quoted from a letter to Jourdain in 1904. Ernst Zermelo [1871-1953] edited Cantor's collected works in 1932, published as Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Cantor's Contributions to the Founding of the Theory of Transfinite Numbers (translated in 1915 by Jourdain) date from 1895 and 1897, and are Cantor's last public presentations of his ideas.

Hallet's book does not contain biographical details on Cantor, and it is not a general history of the rise of set theory. It is foremost an account of Cantor's conception of infinity, both metaphysical and mathematical, and of Cantor's varied presentations, in publications and letters, of his theories of cardinal and ordinal numbers. Hallett focuses tightly on Cantor's writings as he meticulously traces the development of Cantor's ideas, and only after he has established Cantor's views does Hallett include the work of others, unless he gives a joint comparison with the focus on Cantor.

Cantor's set theory is not axiomatic, and although Hallett discusses to some degree, mainly in Part 2, the relationship of the axiomatic theory of sets to Cantor's conception of sets and infinity, the emphasis of the book is on Cantor's work, and only secondarily on the contributions of others as they explored and expanded on Cantor's ideas.

In the first two chapters of Part 2, Hallett is less concerned with how proposed axioms combine to define a theory of sets than with how certain axioms have been discussed and justified (for example, as a means of limiting the comprehension of sets or of representing the iterative conception of sets) and with the claim that the set theoretic paradoxes are meaningfully related to Kant's antinomies of pure reason.

In the final two chapters, Hallett discusses the differing systems of Zermelo and John von Neumann [1903-1957], where his focus in chapter seven is on Zermelo's axiom system in relation to his 1904 and 1908 proofs of the well-ordering theorem, and in chapter eight on von Neumann's theory of ordinals from 1923 (also anticipations of it by others), his use and clarification of the axiom of replacement first proposed by Abraham Fraenkel [1891-1965], and von Neumann's axiomatic theory of functions (not sets) of 1925.

= CONTENTS =

Preface [precedes Contents page]

* Part 1. The Cantorian origins of set theory

Introduction to Part 1: The background to the theory of the ordinals

1. Cantor's theory of infinity

__ 1.1 Free mathematics

__ 1.2 The potential infinite and reductionism

__ 1.3 Cantorian finitism and the concept of set

__ 1.4 Cantor's absolute

2. The ordinal theory of powers

__ 2.1 The generating principles

__ 2.2 The scale of number-classes

__ 2.3 The attack on the continuum problem

___ (a) First step: the uncountability of the continuum and the second number-class

___ (b) What did Cantor achieve?

___ (c) The continuum hypothesis and later developments

3. Cantor's theory of number

__ 3.1 Cantor's abstractionism, set reduction, and Frege-Russell

__ 3.2 Difficulties with the strange theory of 'ones'

__ 3.3 The theory of 'ones' sensibly constructed

__ 3.4 Order-types

__ 3.5 Cantor and well ordering

4. The origin of the limitation of size idea

__ 4.1 The Absolute and limitation of size

__ 4.2 Jourdain's limitation of size theory

__ 4.3 Modifying comprehension by limitation of size

__ 4.4 Mirimanoff

* Part 2. The limitation of size argument and axiomatic set theory

Introduction to Part 2

5. The limitation of size argument

__ 5.1 Fraenkel's argument criticized

__ 5.2 The explanatory role of limitation of size

__ 5.3 The power-set axiom

6. The completability of sets

__ 6.1 The iterative conception

__ 6.2 Completability and Kant's first antinomy

___ (a) Contradiction or sleight of hand?

___ (b) Completability and contructivity

7. The Zermelo system

__ 7.1 Zermelo's separation axiom as a limitation of size principle

__ 7.2 Zermelo's reductionism

___ (a) Zermelo's reductionist treatment of number

___ (b) Zermelo's reductionist treatment of contradiction

__ 7.3 Reductionism and well-ordering

___ (a) Zermelo's 1904 proof

___ (b) Inclusion orderings and the 1908 proof

__ 7.4 The problem of definite properties

8. Von Neumann's reinstatement of the ordinal theory of sets

__ 8.1 The von Neumann theory of ordinals

__ 8.2 The discovery of the replacement axiom

__ 8.3 Limitation of size revisited

Conclusion

Bibliography

Name index

Subject index