The Foundations of Mathematics in the Theory of Sets

Author: John P. Mayberry

Publisher: Cambridge University Press


Category: Mathematics

Page: 424

View: 454

This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

Handbook of Mathematical Logic

Author: J. Barwise

Publisher: Elsevier


Category: Mathematics

Page: 1164

View: 858

The handbook is divided into four parts: model theory, set theory, recursion theory and proof theory. Each of the four parts begins with a short guide to the chapters that follow. Each chapter is written for non-specialists in the field in question. Mathematicians will find that this book provides them with a unique opportunity to apprise themselves of developments in areas other than their own.

Theory of relations

Author: Roland Fraïssé

Publisher: Elsevier Science Ltd


Category: Mathematics

Page: 397

View: 506

The first part of this book concerns the present state of the theory of chains (= total or linear orderings), in connection with some refinements of Ramsey's theorem, due to Galvin and Nash-Williams. This leads to the fundamental Laver's embeddability theorem for scattered chains, using Nash-Williams' better quasi-orderings, barriers and forerunning. The second part (chapters 9 to 12) extends to general relations the main notions and results from order-type theory. An important connection appears with permutation theory (Cameron, Pouzet, Livingstone and Wagner) and with logics (existence criterion of Pouzet-Vaught for saturated relations). The notion of bound of a relation (due to the author) leads to important calculus of thresholds by Frasnay, Hodges, Lachlan and Shelah. The redaction systematically goes back to set-theoretic axioms and precise definitions (such as Tarski's definition for finite sets), so that for each statement it is mentioned either that ZF axioms suffice, or what other axioms are needed (choice, continuum, dependent choice, ultrafilter axiom, etc.).

Foundations of Mathematical Logic

Author: Haskell Brooks Curry

Publisher: Courier Corporation


Category: Mathematics

Page: 408

View: 365

Written by a pioneer of mathematical logic, this comprehensive graduate-level text explores the constructive theory of first-order predicate calculus. It covers formal methods — including algorithms and epitheory — and offers a brief treatment of Markov's approach to algorithms. It also explains elementary facts about lattices and similar algebraic systems. 1963 edition.

Foundations of Set Theory

Author: A.A. Fraenkel

Publisher: Elsevier


Category: Computers

Page: 412

View: 807

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.

Introduction to Logic

and to the Methodology of Deductive Sciences

Author: Alfred Tarski

Publisher: Courier Corporation


Category: Mathematics

Page: 272

View: 919

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

Foundations of Logic, 1903-05

Author: Bertrand Russell

Publisher: Psychology Press


Category: Philosophy

Page: 743

View: 214

This volume covers the period from the beginning of Russell's work on Volume Two of the Principles of Mathematics to the critical discovery of the theory of descriptions in 1905. Foundations of Logic gives a vivid picture of Russell wrestling with the logical paradoxes, often unsuccessfully, as he tries out one foundational scheme after another. This volume provides the key to both Bertrand Russell's philosophy of logic and philosophy of mathematics. It includes unpublished work on the theory of denoting which predates Russell's famous article of 1905 and unpublished manuscripts on the so-called "zig-zag" theory with which Russell attempted to provide a type-free foundation for mathematics. The volume also gathers together for the first time a number of reviews and survey articles, along with two talks on modality and truth. It will be an essential addition to any Bertrand Russell collection.-- Publisher description.

Philosophy of Science, Logic and Mathematics in the Twentieth Century

Author: Stuart Shanker

Publisher: Psychology Press


Category: Philosophy

Page: 504

View: 521

Contents - Introduction. 1. Philosophy of logic 2. Philosophy of mathematics in the 20th century. 3. Frege 4. Wittgenstein's Tractatus 5. Logical postivism 6. The philosophy of physics 7. The philosophy of science 8. Chance, cause and conduct; probability

The Metaphysical Foundations of Logic

Author: Martin Heidegger

Publisher: Indiana University Press


Category: Philosophy

Page: 241

View: 525

Based on a lecture course given by Heidegger at the University of Marburg in the summer of 1928. The first part of the book presents a critique of the thought of Gottfried Wilhelm Leibniz, the seventeenth-century mathematician-scientist-humanist who attempted a synthesis of mathematical physics with the humanistic concerns of the Western European tradition.

Theory and Experiment

Recent Insights and New Perspectives on Their Relation

Author: Diderik Batens

Publisher: Springer Science & Business Media


Category: Science

Page: 296

View: 629

This is not "another collection of contributions on a traditional subject." Even more than we dared to expect during the preparatory stages, the papers in this volume prove that our thinking about science has taken a new turn and has reached a new stage. The progressive destruction of the received view has been a fascinating and healthy experience. At present, the period of destruction is over. A richer and more equilibrated analysis of a number of problems is possible and is being cru'ried out. In this sense, this book comes right on time. We owe a lot to the scholars of the Kuhnian period. They not only did away with obstacles, but in several respects instigated a shift in attention that changed history and philosophy of science in a irreversible way. A c1earcut example - we borrow it from the paper by Risto Hilpinen - concerns the study of science as a process, Rnd not only as a result. Moreover, they apparently reached several lasting results, e.g., concerning the tremendous impact of theoretical conceptions on empirical data. Apart from baffling people for several decades, this insight rules out an other return to simple-minded empiricism in the future.