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Foundations of Set Theory (Volume 67)
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.
412 pages, Hardcover
First published December 1, 1973
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August 28, 2022
Where should one go next after an initial exposure to Cantorian naïve set theory? The technical literature is not only in itself formidable, it is scattered across disparate venues over the span of a century and more. Therefore, anyone aiming for a more mature perspective stands in need of a guide and the present work on the Foundations of Set Theory from the North-Holland Studies in Logic and the Foundations of Mathematics (1984) by some of the foremost authorities in the field, A.A. Fraenkel, Y. Bar-Hillel and A. Levy, comes in very handy. Since this version comprises a second edition of the 1958 first edition by Fraenkel alone, based upon detailed notes left behind after his death in 1966, we shall refer to him as senior author. The interested reader can profit from the perspective of a distinguished expert who himself participated in many of the developments he describes.
What one gets in this edition: a high-level discussion of the ins and outs of modern set theory from the point of view of someone who wants to engage in research into the foundations of mathematics, starting with the famous antinomies first uncovered by Bertrand Russell in 1902. Already in 1895 Cantor and in 1897 Bural-Forti had discovered the paradox that goes under the latter’s name. But as Fraenkel explains, it was Russell’s antinomy that shattered everyone’s confidence that the troublesome problems could be fixed up with a few minor remedies and ushered in a period of crisis. Others – what Fraenkel styles semantical antinomies cropped up before long, due to Richard in 1905 and Grelling and Nelson in 1908.
What attitude should one take on the revision of fundamental concepts these antinomies necessitate? There are two general directions of response: either to take care with the axiomatic foundation or to introduce a theory of types that would bar one from the self-referential constructions that give rise to the contradictions in the first place. The first was taken notably by Zermelo and followed, inter alia, by von Neumann, Bernays and Gödel – including by Fraenkel himself, who early in his career during the 1920’s retouched Zermelo’s axioms from 1908, issuing in what we know today as the Zermelo-Fraenkel set theory and to which chapter two [pp. 15-153] in the present work is devoted. The second approach, viz. type theory, was pursued notably by Russell and Whitehead and later by Quine, subject of chapter three here [pp. 154-209].
Review of contents: Fraenkel’s discussion in chapter two is very meticulous, laying out the theory axiom by axiom and pointing out some of their immediate consequences. Special attention is paid to the axiom of choice and to questions of consistency, independence and well-foundedness. Two further sections round out the chapter, one on unanswered questions (the continuum hypothesis, constructibility and strong infinity) and another on the slightly different approach of von Neumann and Bernays to the axiomatic foundation of set theory, allowing in a consistent way for classes which are more encompassing than sets (in order to deal with the impossibility of defining a set of all sets). The main takeaway of interest to this reviewer is the following: as long as one is willing to entertain well-founded sets only (defined by transfinite induction over the ordinals) everything will be satisfactory; the antinomies go away and one has all one needs for everyday mathematics – i.e., analysis, geometry and algebra. Further speculation on the intricacies of defining sets in larger universes than this can seem otiose.
Chapter three becomes sketchier, perhaps because outside the domain of Fraenkel’s own contributions. Fraenkel has little to say about Russell and Whitehead in the Principia Mathematica but focuses upon the ramified type theory of Quine and others such as Wang and Lorenzen. Also covers non-standard and many-valued logics.
But Fraenkel is not done: chapter four concerns itself with intuitionism, which one may regard as a alternate response to the crisis provoked by the antinomies centered not on restricting the concept of set or its use but on a reform of logic instead. For classical logic permits naïve operations that have to be criticized as soon as one contemplates infinity, or the continuum. Indeed, as Fraenkel states in the historical introduction to this chapter,
Bridging the gap between the domains of discreteness and of continuity, or between arithmetic and geometry, is a central, presumably even the central problem of the foundation of mathematics. Cantor claimed to have bridged the gap, as claimed before by ‘classical’ analysis; the sharpest criticism of these claims has been expressed by the intuitionistic schools. [p. 211]
The main proponent of intuitionism has been L.E.J. Brouwer, hence the second section of chapter four on his constructivism and the third on the fate of the principle of the excluded middle. A nice feature of Fraenkel’s exposition is that he goes into much greater depth than is usual in the literature on the philosophy of mathematics, including section five on choice sequences, spreads and species – Brouwer’s radical suggestions about how we are to do mathematics in practice once unwilling to avail ourselves of suspect transfinite operations.
Chapter five [pp. 275-345] then takes up the remaining option in response to the crisis in foundations, Hilbert’s formalist program and the questions of consistency, completeness, categoricalness, independence, decidability and recursiveness. After laying out the basics of formal systems and model theory, in section seven he reviews the limits imposed upon Hilbert’s goals by the theorems of Gödel, Tarski and Church. The chapter concludes with two sections on metamathematics, semantics and philosophical remarks.
Summary: numerous solicisms or non-idiomatic turns of phrase betray that the authors must not have been native speakers of English. There are no hard derivations but many informal expositions. Frankel’s text should prove good for 1) an overview of what has been done and why; 2) precise statements of results one will have heard of before but not seen spelled out in exacting detail (e.g. Cohen showed independence of the axiom of choice relative to ZF in 1963 etc.); 3) near-complete references to the original literature up to the time of publication of the second edition in 1984. Think of not as a textbook but as a book-length bibliographical essay. For instance, the discussion of Brouwerianism in chapter four is much more complete than anything else readily available in English of Brouwer’s original work. Four stars, in view of its exceptional clarity.
For further reading (neither of which has this recensionist had the opportunity as yet to consult thoroughly): Ebbinghouse, Flum and Thomas, Mathematical Logic: a less sophisticated discourse than Fraenkel’s but rich in concrete results and their derivations – supposedly intended for undergraduates!; Jech, Set Theory: a sprawling tome that covers just about everything in exhaustive technical detail up to and including large cardinals and forcing.
What one gets in this edition: a high-level discussion of the ins and outs of modern set theory from the point of view of someone who wants to engage in research into the foundations of mathematics, starting with the famous antinomies first uncovered by Bertrand Russell in 1902. Already in 1895 Cantor and in 1897 Bural-Forti had discovered the paradox that goes under the latter’s name. But as Fraenkel explains, it was Russell’s antinomy that shattered everyone’s confidence that the troublesome problems could be fixed up with a few minor remedies and ushered in a period of crisis. Others – what Fraenkel styles semantical antinomies cropped up before long, due to Richard in 1905 and Grelling and Nelson in 1908.
What attitude should one take on the revision of fundamental concepts these antinomies necessitate? There are two general directions of response: either to take care with the axiomatic foundation or to introduce a theory of types that would bar one from the self-referential constructions that give rise to the contradictions in the first place. The first was taken notably by Zermelo and followed, inter alia, by von Neumann, Bernays and Gödel – including by Fraenkel himself, who early in his career during the 1920’s retouched Zermelo’s axioms from 1908, issuing in what we know today as the Zermelo-Fraenkel set theory and to which chapter two [pp. 15-153] in the present work is devoted. The second approach, viz. type theory, was pursued notably by Russell and Whitehead and later by Quine, subject of chapter three here [pp. 154-209].
Review of contents: Fraenkel’s discussion in chapter two is very meticulous, laying out the theory axiom by axiom and pointing out some of their immediate consequences. Special attention is paid to the axiom of choice and to questions of consistency, independence and well-foundedness. Two further sections round out the chapter, one on unanswered questions (the continuum hypothesis, constructibility and strong infinity) and another on the slightly different approach of von Neumann and Bernays to the axiomatic foundation of set theory, allowing in a consistent way for classes which are more encompassing than sets (in order to deal with the impossibility of defining a set of all sets). The main takeaway of interest to this reviewer is the following: as long as one is willing to entertain well-founded sets only (defined by transfinite induction over the ordinals) everything will be satisfactory; the antinomies go away and one has all one needs for everyday mathematics – i.e., analysis, geometry and algebra. Further speculation on the intricacies of defining sets in larger universes than this can seem otiose.
Chapter three becomes sketchier, perhaps because outside the domain of Fraenkel’s own contributions. Fraenkel has little to say about Russell and Whitehead in the Principia Mathematica but focuses upon the ramified type theory of Quine and others such as Wang and Lorenzen. Also covers non-standard and many-valued logics.
But Fraenkel is not done: chapter four concerns itself with intuitionism, which one may regard as a alternate response to the crisis provoked by the antinomies centered not on restricting the concept of set or its use but on a reform of logic instead. For classical logic permits naïve operations that have to be criticized as soon as one contemplates infinity, or the continuum. Indeed, as Fraenkel states in the historical introduction to this chapter,
Bridging the gap between the domains of discreteness and of continuity, or between arithmetic and geometry, is a central, presumably even the central problem of the foundation of mathematics. Cantor claimed to have bridged the gap, as claimed before by ‘classical’ analysis; the sharpest criticism of these claims has been expressed by the intuitionistic schools. [p. 211]
The main proponent of intuitionism has been L.E.J. Brouwer, hence the second section of chapter four on his constructivism and the third on the fate of the principle of the excluded middle. A nice feature of Fraenkel’s exposition is that he goes into much greater depth than is usual in the literature on the philosophy of mathematics, including section five on choice sequences, spreads and species – Brouwer’s radical suggestions about how we are to do mathematics in practice once unwilling to avail ourselves of suspect transfinite operations.
Chapter five [pp. 275-345] then takes up the remaining option in response to the crisis in foundations, Hilbert’s formalist program and the questions of consistency, completeness, categoricalness, independence, decidability and recursiveness. After laying out the basics of formal systems and model theory, in section seven he reviews the limits imposed upon Hilbert’s goals by the theorems of Gödel, Tarski and Church. The chapter concludes with two sections on metamathematics, semantics and philosophical remarks.
Summary: numerous solicisms or non-idiomatic turns of phrase betray that the authors must not have been native speakers of English. There are no hard derivations but many informal expositions. Frankel’s text should prove good for 1) an overview of what has been done and why; 2) precise statements of results one will have heard of before but not seen spelled out in exacting detail (e.g. Cohen showed independence of the axiom of choice relative to ZF in 1963 etc.); 3) near-complete references to the original literature up to the time of publication of the second edition in 1984. Think of not as a textbook but as a book-length bibliographical essay. For instance, the discussion of Brouwerianism in chapter four is much more complete than anything else readily available in English of Brouwer’s original work. Four stars, in view of its exceptional clarity.
For further reading (neither of which has this recensionist had the opportunity as yet to consult thoroughly): Ebbinghouse, Flum and Thomas, Mathematical Logic: a less sophisticated discourse than Fraenkel’s but rich in concrete results and their derivations – supposedly intended for undergraduates!; Jech, Set Theory: a sprawling tome that covers just about everything in exhaustive technical detail up to and including large cardinals and forcing.
August 24, 2008
I was studying sets and logic, and i found that this book was really help me out.
Displaying 1 - 2 of 2 reviews