This is a set of three volumes containing edited versions of papers and commentaries presented in invited symposium sessions of the Eighth World Congress of the Econometric Society. The papers summarize and interpret recent key developments and discuss future directions in a wide range of topics in economics and econometrics. The papers cover both theory and applications. Written by leading specialists in their fields, these volumes provide a unique survey of progress in the discipline.

Quasi-set theory by S. French and D. Krause has been so far the most promising attempt of a formal theory of non-individuals. However, due to its sharp bivalent truth valuations, maximally fine-grained binary relations are readily found, in which members of equivalence classes are substitutable for each other in formulas salva veritate. Hence its mentioning and non-mentioning of individuals differs from existing set theory with defined identity merely by the range of nominal definitions. On a semantic level, (...) quasi-set theory does not provide an interpretation with explanatory power of its language terms as non-individuals, and it is not easy to see how such an interpretation can be set up. (shrink)

Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...) the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond. (shrink)

LN , so f lies in the elementary submodel M'. Clearly co 9 M' . It follows that 6 = {f(n): n em} is included in M'. Hence the ordinals of M' form an initial ...

The rough set theory has interesting properties such as that a rough set is considered as distinct sets in distinct knowledge bases, and that distinct rough sets are considered as one same set in a certain knowledge base. This leads to a significant philosophical interpretation: a concept (or phenomenon) may be understood as different ones in different philosophical perspectives, while different concepts (or phenomena) may be understood as a same one in a certain philosophical perspective. Such properties of rough (...) set theory produce a mathematical model to support critical realism and the theory ladenness of observation in the philosophy of science. (shrink)

This three volume set contains papers presented at the Seventh World Congress of the Econometric Society. The first volume contains three papers presented at the Seventh World Congress of the Econometric Society which summarize and interpret key recent developments and discuss current and future directions in a wide range of topics in economics and econometrics. They cover both theory and applications. Authored by leading specialists in their fields, these volumes provide a unique survey of progress in the discipline. The (...) second volume comprising three papers which examine the latest developments in economic theory, applied economics and econometrics presented at the Seventh World Congress of the Econometric Society in Tokyo in August 1995. The topics were carefully selected to represent the most active fields in the discipline over the past five years. Written by the leading authorities in their fields, each paper provides a unique survey of the current state of knowledge in economics. Designed to make the material accessible to a general audience of economists, these volumes should be helpful to anyone with a good undergraduate training in economics who wishes to follow new ideas and tendencies in the subject. The third of three volumes contains papers presented at the Seventh World Congress of the Econometric Society. The papers summarize and interpret key recent developments and discuss current and future directions in a wide range of topics in economics and econometrics. They cover both theory and applications. Authored by leading specialists in their fields, these volumes provide a unique survey of progress in the discipline. (shrink)

Public sphere is an important idea of Habermas in the early research, which guided his latter research, especially in political philosophy field. According to Habermas’ research on public sphere, this paper researches public sphere’s significance in solving the legalization crisis of capitalism and remedying the democratic theory of bourgeoisie. Public sphere idea set up a new model of the democratic theory, deliberative democracy, which is better than democracy of both liberalism and republicanism, and become the most important theme (...) of Habermas in theory and practice. (shrink)

The Fifth International Congress of Logic, Methodology and Philosophy of Science was held at the University of Western Ontario, London, Canada, 27 August to 2 September 1975. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, and was sponsored by the National Research Council of Canada and the University of Western Ontario. As those associated closely with the work of the Division over the years (...) know well, the work undertaken by its members varies greatly and spans a number of fields not always obviously related. In addition, the volume of work done by first rate scholars and scientists in the various fields of the Division has risen enormously. For these and related reasons it seemed to the editors chosen by the Divisional officers that the usual format of publishing the proceedings of the Congress be abandoned in favour of a somewhat more flexible, and hopefully acceptable, method of pre sentation. Accordingly, the work of the invited participants to the Congress has been divided into four volumes appearing in the University of Western Ontario Series in Philosophy of Science. The volumes are entitled, Logic, Foundations of Mathematics and Computability Theory, Foun dational Problems in the Special Sciences, Basic Problems in Methodol ogy and Linguistics, and Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science. (shrink)

Bonjour (1985: 101 and 1999: 124) and other coherence theorists of justification before him (e.g. Ewing, 1934: 246) have complained that we do not have a satisfactory analysis of the notion of coherence. The problem with existing accounts of coherence is that they try to bring precision to our intuitive notion of coherence independently of the particular role that it is meant to play within the coherence theory of justification (e.g Lewis, 1946: 338). This is a mistake: it does (...) not make any sense to ask what precisely makes for a more coherent information set independently of the particular role that coherence is supposed to play within the context in question. What is this context and what is this role? The coherence theory of justification rides on a particular common sense intuition: when we gather information from less than fully reliable sources, then the more coherent the story that materializes is, the more confident we may be, ceteris paribus. Within the context of information gathering from certain types of sources, coherence is a property of stories which plays a confidence boosting role. But what features should the information sources have, so that the coherence of the information set is indeed a determinant of our degree of confidence in question? And what goes into the ceteris paribus clause? In other words, what other factors affect our confidence in the information set in question? (shrink)

This collection is the final volume of a four book survey of the state of phenomenology fifty years after the death of Edmund Husserl. Its publication represents a landmark in the comprehensive treatment of contemporary phenomenology in all its vastness and richness. The diversity of the issues raised here is dazzling, but the main themes of Husserl's thought are all either explicitly treated, or else they underlie the ingenious approaches found here. Time, historicity, intentionality, eidos, meaning, possibility/reality, and teleology are (...) the main concerns of this collection devoted to studies in aesthetics, metaphysics and literary interpretation, written by such authors as, among others, R. Cobb-Stevens, C. Moreno Marquez, J. Swiecimski, Sitansu Ray and M. Kronegger. These original studies of phenomenological aesthetics and literary theory by scholars from all parts of the world were gathered by the World Institute for Advanced Phenomenological Research and Learn ing during the year 1988/89 during its assessment of the phenomeno logical movement, fifty years after Husserl's death. IX A -T. Tymieniecka (ed.), Analecta Husserliana, Vol. XXXVII, ix. (shrink)

Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematical knowledge. A (...) third is to serve mathematics, by lending insight into the various fields and suggesting fruitful techniques of research. A fourth purpose of a foundation is to provide an arena for exploring relations and interactions between mathematical fields. While set theory does better with regard to some of these and worse with regard to others, it has become the de facto arena for deciding questions of existence, something one might expect of a foundation. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

GENERALIZED RECUBION THEORY II © North-Holland Publishing Company (1978) MONOTONE QUANTIFIERS AND ADMISSIBLE SETS Ion Barwise University of Wisconsin ...

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). -/- By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive (...) way of fleshing out this idea is synonymous with a ZF-like theory. -/- This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974). (shrink)

In this contribution, I explore the possibility of characterizing the emergence of time in causal set theory (CST) in terms of the growing block universe (GBU) metaphysics. I show that although GBU seems to be the most intuitive time metaphysics for CST, it leaves us with a number of interpretation problems, independently of which dynamics we choose to favor for the theory —here I shall consider the Classical Sequential Growth and the Covariant model. Discrete general covariance of the (...) CSG dynamics does not allow us to individuate a single history of the universe (defined by a causal history of different causal sets), thereby making the claim that ‘the past exists’ at best problematic. In addition, because the evolution of the universe in CSG dynamics leads to an outward branching causal tree, it becomes impossible to determine a proper ‘line of becoming’, thereby blurring the presentists’ claim that only the present exists. Similarly, the covariant approach runs into the same, if not even more severe problems, since each configuration of the universe would amount to a set of possible causal sets, thereby making the individuation of a single configuration of the universe —and thus the physical interpretation of the theory— implausible. (shrink)

This chapter surveys set theory, model theory, and computability theory: how they first emerged from the foundations of mathematics, and how they have developed since. There are any amounts of mathematical technicalities in the background, but the chapter highlights those themes that have some philosophical resonance.

The ArgumentThis paper is about the context of Albert Einstein's concerns at the time of a most intense intellectual effort — his own and that of a small group of scientists concerned with classical quantum theory. I describe contemporaneous interactions and differing views about the prospects for and the significance of the First Solvay Congress of 1911 as voiced by major participants. There are two axes around which the paper evolves: the Einstein-Nernst-Lorentz dialogue and the public institutional creation of (...) the “Solvay” stage. In certain ways, this paper is about personal and institutional patronage: the working out of a difficult theoretical impasse requires individual and collective moral support. It is about the forging of the identity of a scientific problem and the personal and institutional setting, the public space, in which this problem was collectively addressed. But I also uncover heuristic playfulness and flexibility which accompanies theoretical change. (shrink)

This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject.

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes (...) that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 379 the minister of a very influential and liberal congregation. In 1860 he began publication in Cincinnati of The Dial, successor to the New England transcendentalist journal, and used its pages to promote religious liberalism, philosophical transcendentalism, and social reform. In 1863 he went to London where he became the head of the Ethical Society. Under the influence of Feuerbach and "left-Hegelians" he travelled widely in the (...) interests of "free thought." The most amazing of these four, however, is August Willich: an East Prussian soldier, ready to fight again to restore the 1848 revolution in Germany, discouraged by Marx from doing so, emigrating to Cincinnati to raise funds among the German 1848 refugees there and urging them to continue the lost cause in Germany, then wholeheartedly fighting for the Union Cause as a General in the Civil War, and finally joining the Populist revolt. As Easton points out, these four all illustrate the close relation between Hegel and the spirit of 1848. In addition to the stories of these men and their ideals, the volume contains a hundred pages of excellent specimens of their writngs, and copious references to obscure source-materials. The book is both a valuable reference work and an entertaining set of short stories. I--IERBERTW. SCHNEIDER Claremont, CaliJornia Theory of. Man. By Francisco Romero. Translated from the Spanish by William F. Cooper, with an Introduction by William J. Kflgore. (Berkeley and Los Angeles: University of California Press, 1964. Pp. xxix + 324. $8.00.) The late Argentinian philosopher Francisco Romero, who died unexpectedly in 1962 after a brilliant career of more than thirty years of teaching and writing, was by common acclaim of his fellow-philosophers in Latin America their most distinguished philosopher in recent years. He received many honors and distinctions both at home and abroad, including the grand award of honor of the Argentinian Society of Writers. The most eminent philosophers in Latin American prepared a notable Festschri]t in honor of his 70th birthday, whose publication was unfortunately delayed and, therefore, could not be presented to him personally in his life time.1 Franci~o Romero was in fact as well as in acclaim the "Dean of Latin American Philosophy." It was high time that his central work Teorga del Hombre 2 should be translated. It is his magnum opus and represents the most comprehensive and systematic development of his philosophy. Romero was fortunate in having as his translator William F. Cooper who knew Romero personally, and was intimately acquainted with his philosophy. In addition, he had the unusual privilege of doing much of the translation in Romero's own library in Buenos Aires. In short, the translation is a labor of love and esteem.8 I myself had the privilege of coming to know Professor Romero in 1943, when as secretarygeneral of our Philosophical Association I was commissioned to make contact with our Latin American fellow-philosophers with a view to exploring the possibility of organizing InterAmerican congresses of philosophy and of forming an Inter-American society of philosophy, both of which projects, in due season, were accomplished. Romero welcomed these undertakings with characteristic enthusiasm and offered his fullest cooperation and support. He had always been a most active and indefatigable promoter of better understanding between North and South America and, especially, of closer cooperation between the philosophers of 1Homenaje a Francisco Romero, Faculdad de Filosofia y Letras, Universidad de Buenos Aires. Editorial Losada, Buenos Aires, 1952. 8In further attestation of Professor Cooper's special qualification for this translation it should be mentioned that he prepared for the Homenaje, already referred to, the most complete, up-to-date bibliography of Romero's extensive publications as well as an inventory of Romero's many works still in progress at the time of his death. 380 HISTORY OF PHILOSOPHY the two Americas. All members of our association who came to know him in Buenos Aires were warmly welcomed, and no doubt also experienced, as I did, the warm and generous hospitality of Professor Romero and his wife in their home in Martinez. Francisco Romero was not only professor of philosophy at the Universities of Buenos Aires and La Plato, but was also... (shrink)

The emphasis in this collection is clearly on logic, and this is one reason why it lacks the overall diversity and richness of the 1960 Stanford volume. However, the eight sections do contain much interesting material; in the mathematical logic section Kochen and Specker continue their study of logics appropriate for quantum theory, Vaught presents several new results about the Löwenheim-Skolem theorem, and Büchi studies second-order ordinal theory from the viewpoint of automata theory; the section on foundations (...) of mathematical theories contains papers on higher-order logic by Kaplan and Montague, model theory and ultra-products by Keisler, definability in set theory by Lévy. The topic of the justification of formal theories was the essential topic of the philosophy of logic section: an especially interesting paper is that of A. Robinson in which the present state of Hilbertean Formalism is analyzed, and in which the notion of potential truth is formalized. The section on the philosophy of science contains a single paper, by Hesse, on metaphor and explanation; Braithwaite, Hintikka, Jeffrey, and Kyburg contribute to a lively section on probability and induction; the section on methodology in physical science takes as topics the theory of relativity, causality, and irreversibility. The last two sections, on the philosophy of the life sciences, and on history of logic and philosophy of science, contain, respectively, papers by Davidson and Suppes on meaning and concept formation, and essays by Church and Geach on existential import historically considered and intentionality among the medievals. The collection would have been better if more of the contributed papers had been included.—P. J. M. (shrink)

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...) defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, (...) a space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted. (shrink)

This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations.

Although political theory was pronounced dead only a few short years ago, this collection of articles shows that much life is left in contemporary political theory. Based on a symposium concerning human rights held at the Sixth World Congress of the International Political Science Association held at Geneva in 1964, the collection includes papers by Macpherson, Polin, Chapman, Cranston, Raphael, Mayo, Schneider, and Fawcett. Macpherson and Polin set the context by exploring the concept of rights in Hobbes and (...) Locke. While the other papers have an eye on traditional discussions, they are also concerned with exploring what "rights" does and can mean in the contemporary world. The international gathering of authors brings together diverse points of view on the common problem of human rights.--R. J. B. (shrink)

In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or (...) that neither is. If there is reason to accept the view that the set-theoretic universe is open-ended, that will be because such a view is the most compelling one to adopt on the purely ontological front. (shrink)

This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.

In a recent series of papers, I and others have advanced new logical approaches to familiar paradoxes. The key to these approaches is to accept full classical logic, and to accept the principles that cause paradox, while preventing trouble by allowing a certain sort ofnontransitivity. Earlier papers have treated paradoxes of truth and vagueness. The present paper will begin to extend the approach to deal with the familiar paradoxes arising in naive set theory, pointing out some of the promises (...) and pitfalls of such an approach. (shrink)

The book discusses the fate of universality and a universal set in several set theories.

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 379 the minister of a very influential and liberal congregation. In 1860 he began publication in Cincinnati of The Dial, successor to the New England transcendentalist journal, and used its pages to promote religious liberalism, philosophical transcendentalism, and social reform. In 1863 he went to London where he became the head of the Ethical Society. Under the influence of Feuerbach and "left-Hegelians" he travelled widely in the (...) interests of "free thought." The most amazing of these four, however, is August Willich: an East Prussian soldier, ready to fight again to restore the 1848 revolution in Germany, discouraged by Marx from doing so, emigrating to Cincinnati to raise funds among the German 1848 refugees there and urging them to continue the lost cause in Germany, then wholeheartedly fighting for the Union Cause as a General in the Civil War, and finally joining the Populist revolt. As Easton points out, these four all illustrate the close relation between Hegel and the spirit of 1848. In addition to the stories of these men and their ideals, the volume contains a hundred pages of excellent specimens of their writngs, and copious references to obscure source-materials. The book is both a valuable reference work and an entertaining set of short stories. I--IERBERTW. SCHNEIDER Claremont, CaliJornia Theory of. Man. By Francisco Romero. Translated from the Spanish by William F. Cooper, with an Introduction by William J. Kflgore. (Berkeley and Los Angeles: University of California Press, 1964. Pp. xxix + 324. $8.00.) The late Argentinian philosopher Francisco Romero, who died unexpectedly in 1962 after a brilliant career of more than thirty years of teaching and writing, was by common acclaim of his fellow-philosophers in Latin America their most distinguished philosopher in recent years. He received many honors and distinctions both at home and abroad, including the grand award of honor of the Argentinian Society of Writers. The most eminent philosophers in Latin American prepared a notable Festschri]t in honor of his 70th birthday, whose publication was unfortunately delayed and, therefore, could not be presented to him personally in his life time.1 Franci~o Romero was in fact as well as in acclaim the "Dean of Latin American Philosophy." It was high time that his central work Teorga del Hombre 2 should be translated. It is his magnum opus and represents the most comprehensive and systematic development of his philosophy. Romero was fortunate in having as his translator William F. Cooper who knew Romero personally, and was intimately acquainted with his philosophy. In addition, he had the unusual privilege of doing much of the translation in Romero's own library in Buenos Aires. In short, the translation is a labor of love and esteem.8 I myself had the privilege of coming to know Professor Romero in 1943, when as secretarygeneral of our Philosophical Association I was commissioned to make contact with our Latin American fellow-philosophers with a view to exploring the possibility of organizing InterAmerican congresses of philosophy and of forming an Inter-American society of philosophy, both of which projects, in due season, were accomplished. Romero welcomed these undertakings with characteristic enthusiasm and offered his fullest cooperation and support. He had always been a most active and indefatigable promoter of better understanding between North and South America and, especially, of closer cooperation between the philosophers of 1Homenaje a Francisco Romero, Faculdad de Filosofia y Letras, Universidad de Buenos Aires. Editorial Losada, Buenos Aires, 1952. 8In further attestation of Professor Cooper's special qualification for this translation it should be mentioned that he prepared for the Homenaje, already referred to, the most complete, up-to-date bibliography of Romero's extensive publications as well as an inventory of Romero's many works still in progress at the time of his death. 380 HISTORY OF PHILOSOPHY the two Americas. All members of our association who came to know him in Buenos Aires were warmly welcomed, and no doubt also experienced, as I did, the warm and generous hospitality of Professor Romero and his wife in their home in Martinez. Francisco Romero was not only professor of philosophy at the Universities of Buenos Aires and La Plato, but was also... (shrink)

Naive set theory, as found in Frege and Russell, is almost universally believed to have been shown to be false by the set-theoretic paradoxes. The standard response has been to rank sets into one or other hierarchy. However it is extremely difficult to characterise the nature of any such hierarchy without falling into antinomies as severe as the set-theoretic paradoxes themselves. Various attempts to surmount this problem are examined and criticised. It is argued that the rejection of naive set (...) theory inevitably leads one into a severe scepticism with regard to the feasibility of giving a systematic semantics for set theory. It is further argued that this is not just a problem for philosophers of mathematics. Semantic scepticism in set theory will almost inevitably spill over into total pessimism regarding the prospects for an explanatory theory of language and meaning in general. The conclusion is that those who wish to avoid such intellectual defeatism need to look seriously at the possibility that it is the logic used in the derivation of the paradoxes, and not the naive set theory itself, which is at fault. (shrink)

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also (...) explored. In Chapter 3, I investigate forcing with urelements and develop a new approach that addresses a drawback of the existing machinery. I demonstrate that forcing can preserve, destroy, and recover the axioms isolated in Chapter 2 and discuss how Boolean ultrapowers can be applied in urelement set theory. Chapter 4 delves into class theory with urelements. I first discuss the issue of axiomatizing urelement class theory and then explore the second-order reflection principle with urelements. In particular, assuming large cardinals, I construct a model of second-order reflection where the principle of limitation of size fails. (shrink)

In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. (...) Berry further defends her approach against a number of possible objections, and she shows how a notion of logical possibility that is useful in formulating Potentialist set theory connects in important ways with philosophy of language, metametaphysics and philosophy of science. Her book will appeal to readers with interests in the philosophy of set theory, modal logic, and the role of mathematics in the sciences. (shrink)

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.