David Hilbert famously remarked, “No one will drive us from the paradise that Cantor has created.” This volume offers a guided tour of modern mathematics’ Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor’s transfinite paradise; axiomatic set theory; logical objects and logical types; independence results and the universe of sets; and the constructs and reality of mathematical structure. Philosophers (...) and mathematicians will find an abundance of intriguing topics in this text, which is appropriate for undergraduate- and graduate-level courses. 1989 ed. 32 figures. (shrink)

We show that the following conjecture about the universe V of all sets is wrong: for all set-theoretical (i.e., first order) schemata true in V there is a transitive set "reflecting" in such a way that the second order statement corresponding to is true in . More generally, we indicate the ontological commitments of any theory that exploits reflection principles in order to yield large cardinals. The disappointing conclusion will be that our only apparently good arguments for the existence (...) of large cardinals have bad presuppositions. (shrink)

Quine's set theory, New Foundations, has often been treated as an anomaly in the history and philosophy of set theory. In this book, Sean Morris shows that it is in fact well-motivated, emerging in a natural way from the early development of set theory. Morris introduces and explores the notion of set theory as explication: the view that there is no single correct axiomatization of set theory, but rather that the various axiomatizations all serve (...) to explicate the notion of set and are judged largely according to pragmatic criteria. Morris also brings out the important interplay between New Foundations, Quine's philosophy of set theory, and his philosophy more generally. We see that his early technical work in logic foreshadows his later famed naturalism, with his philosophy of set theory playing a crucial role in his primary philosophical project of clarifying our conceptual scheme and specifically its logical and mathematical components. (shrink)

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. (shrink)

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account (...) of the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and (...) to computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

This book has two main goals: first, to show that Quine's New Foundations set theory is better motivated than often assumed; and second, to defend Quine's philosophy of set theory. It is divided into three parts. The first concerns the history of set theory and argues against readings that see the iterative conception of set being the dominant notion of set from the very beginning. The second part concerns Quine's philosophy of set theory. Part (...) 3 is a contemporary assessment of the philosophical status of NF. Here one of the central targets is Boolos's defense of the iterative conception of set and his dismissal of NF as completely unnatural.As this is a very short review, I will avoid specific points and... (shrink)

In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that this concept (...) is given to us with a certain sense as the objective focus of a phenomenologically reduced intentional experience. The concept of set that ZF describes, I claim, is that of a multiplicity of coexisting elements that can, as a consequence, be a member of another multiplicity. A set is conceived as a quantitatively determined collection of objects that is, by necessity, ontologically dependent on its elements, which, on the other hand, must exist independently of it. A close scrutiny of the essential characters of this conception seems to be sufficient to ground the set-theoretic hierarchy and the axioms of ZF. (shrink)

In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that this concept (...) is given to us with a certain sense as the objective focus of a ”phenomenologically reduced“ intentional experience.The concept of set that ZF describes, I claim, is that of a multiplicity of coexisting elements that can, as a consequence, be a member of another multiplicity. A set is conceived as a quantitatively determined collection of objects that is, by necessity, ontologically dependent on its elements, which, on the other hand, must exist independently of it. A close scrutiny of the essential characters of this conception seems to be sufficient to ground the set-theoretic hierarchy and the axioms of ZF. (shrink)

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)

The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of (...) symbols אₒ and c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics. (shrink)

The study of logic goes back more than two thousand years and in that time many symbols and diagrams have been devised. Around 300 BC Aristotle introduced letters as term-variables, a "new and epoch-making device in logical technique." (W. & M. Kneale The Development of Logic (1962, p. 61). The modern era of mathematical notation in logic began with George Boole (1815- 1864), although none of his notation survives. Set theory came into being in the late 19th and early (...) 20th centuries, largely a creation of Georg Cantor (1845-1918). See MacTutor's A history of set theory or, for more detail, Set theory from the Stanford Encyclopedia of Philosophy. (shrink)

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph (...) conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. (shrink)

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

Until recently, the paradoxes of set theory were hardly used at all in the analysis of dialectical contradiction. "Violations of the Aristotelian law of contradiction have been found everywhere except where logic and mathematics saw them." Today a practice of study of paradoxes in set theory by the devices of materialist dialectics is taking shape in our literature.

The ten contributions in this volume range widely over topics in the philosophy of mathematics. The four papers in Part I (entitled "Set Theory, Inconsistency, and Measuring Theories") take up topics ranging from proposed resolutions to the paradoxes of naïve set theory, paraconsistent logics as applied to the early infinitesimal calculus, the notion of "purity of method" in the proof of mathematical results, and a reconstruction of Peano's axiom that no two distinct numbers have the same successor. (...) Papers in the second part ("The Challenge of Nominalism") concern the nominalistic thesis that there are no abstract objects. The two contributions in Part III ("Historical Background") consider the contributions of Mill, Frege, and Descartes to the philosophy of mathematics. (shrink)

Ernst Cassirer occupies a unique space in Twentieth-century philosophy. A great liberal humanist, his multi-faceted work spans the history of philosophy, the philosophy of science, intellectual history, aesthetics, epistemology, the study of language and myth, and more. Cassirer's thought also anticipates the renewed interest in the origins of analytic and continental philosophy in the Twentieth Century and the divergent paths taken by the 'logicist' and existential traditions, epitomised by his now legendary debate in 1929 with the (...) philosopher Martin Heidegger, over the question "What is the Human Being?" The Philosophy of Symbolic Forms is Cassirer's most important work. It was first published in German in 1923, the third and final volume appearing in 1929. In it Cassirer presents a radical new philosophical worldview - at once rich, creative and controversial - of human beings as fundamentally "symbolic animals," placing signs and systems of expression between themselves and the world. This major new translation of all three volumes, the first for over fifty years, brings Cassirer's magnum opus to a new generation of students and scholars. Taken together, the three volumes of The Philosophy of Symbolic Forms are a vital treatise on human beings as symbolic animals and a monumental expression of neo-Kantian thought. Correcting important errors in previous English editions, this translation reflects the contributions of significant advances in Cassirer scholarship over the last twenty to thirty years. Each volume includes a new introduction and translator's notes by S. G. Lofts, a foreword by Peter Gordon, a glossary of key terms, and a thorough index. (shrink)

Alain Badiou's Being and Event continues to impact philosophical investigations into the question of Being. By exploring the central role set theory plays in this influential work, Burhanuddin Baki presents the first extended study of Badiou's use of mathematics in Being and Event. Adopting a clear, straightforward approach, Baki gathers together and explains the technical details of the relevant high-level mathematics in Being and Event. He examines Badiou's philosophical framework in close detail, showing exactly how it is 'conditioned' by (...) the technical mathematics. Clarifying the relevant details of Badiou's mathematics, Baki looks at the four core topics Badiou employs from set theory: the formal axiomatic system of ZFC; cardinal and ordinal numbers; Kurt G. (shrink)

This book presents the current feminist critique of science and the philosophy of science in such a way that students of philosophy of science, philosophers, feminist theorists, and scientists will find the material accessible and intellectually rigorous.Contemporary feminist debate, as well as the debate brought on by the radical critics of science, assumes—incorrectly—that certain movements in philosophy of science and science-driven theory are understood in their dynamics as well as in their details. All too often, labels (...) such as “Kuhnian” or “positivistic” are taken for granted, and much of the contemporary postmodern or post-structuralist feminist theory that sets out to criticize science does little to alleviate the reader’s lack of knowledge with regard to such movements.Unlike other texts, Philosophies of Science: Feminist Theories provides a student-oriented framework so that, for example, positivism is given a thorough grounding before the feminist critique of such epistemological theory is given. Other movements discussed include the Kuhnian turn, sociology of science, and the radical critique of science. Feminist theory and critique are interwoven throughout, with one chapter devoted to feminist thought, which includes the work of such thinkers as Longino, Hararway, Hubbard, Nelson, Harding, and Keller. (shrink)

The first in-depth reference in the field that combines scientific knowledge with philosophical inquiry, _The Philosophy of Science: An Encyclopedia_ is a two-volume set that brings together an international team of leading scholars to provide over 130 entries on the essential concepts in the philosophy of science. _The areas covered include:_ biology chemistry epistemology and metaphysics physics psychology and mind the social sciences key figures in the combined studies of science and philosophy. The essays represent the most (...) up-to-date philosophical thinking on timeless scientific topics such as: determinism, explanation, laws of nature, perception, individuality, time, and economics as well as timely topics like adaptation, conservation biology, quantum logic, consciousness, evolutionary psychology, and game theory. (shrink)

Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James (...) H. Schmerl; 9. History of constructivism in the 20th century A. S. Troelstra; 10. A very short history of ultrafinitism Rose M. Cherubin and Mirco A. Mannucci; 11. Sue Toledo's notes of her conversations with Gödel in 1972-1975 Sue Toledo; 12. Stanley Tennenbaum's Socrates Curtis Franks; 13. Tennenbaum's proof of the irrationality of [the square root of] 2́. (shrink)

ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.

We investigate the fundamental relationship between philosophical aesthetics and the philosophy of nature, arguing for a position in which the latter encompasses the former. Two traditions are set against each other, one is natural aesthetics, whose covering philosophy is Idealism, and the other is the aesthetics of nature, the position defended in this article, with the general program of a comprehensive philosophy of nature as its covering theory. Our approach is philosophical, operating within the framework of (...) the ontology of the process of the production of art, especially the views of Antonin Artaud, Heidegger, Bakhtin, Deleuze, and Guattari. We interrogate Dilthey and Worringer while outlining an ontology of art based on the production of nonhuman images and a nonpersonal experiential field of nature. (shrink)

Standard presentations of axioms for set theory as truths simpliciter about actual-objects the sets-confront a number of puzzles associated with platonism and foundationalism. In his classic, Zermelo suggested an alternative "many worlds" view. Independently, Putnam proposed something similar, explicitly incorporating modality. A modal-structural synthesis of these ideas is sketched in which obstacles to their formalization are overcome. Extendability principles are formulated and used to motivate many small large cardinals. The use of second-order logic as a coherent and clear framework (...) for set theory is supported. (shrink)