Set Theory and the Continuum Problem

It also presents, for the first time in a textbook, the double induction and superinduction principles, and Cowen's theorem. The book will interest students and researchers in logic and set theory.

Author: Raymond M. Smullyan

Publisher: Oxford University Press, USA

ISBN: UOM:39015041010680

Category: Mathematics

Page: 288

View: 389

Set Theory and the Continuum Problem is a novel introduction to set theory, including axiomatic development, consistency, and independence results. It is self-contained and covers all the set theory that a mathematician should know. Part I introduces set theory, including basic axioms, development of the natural number system, Zorn's Lemma and other maximal principles. Part II proves the consistency of the continuum hypothesis and the axiom of choice, with material on collapsing mappings, model-theoretic results, and constructible sets. Part III presents a version of Cohen's proofs of the independence of the continuum hypothesis and the axiom of choice. It also presents, for the first time in a textbook, the double induction and superinduction principles, and Cowen's theorem. The book will interest students and researchers in logic and set theory.
1996

Set Theory and the Continuum Hypothesis

This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

Author: Paul J. Cohen

Publisher: Courier Corporation

ISBN: 9780486469218

Category: Mathematics

Page: 154

View: 414

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.
2008-12-09

Set Theory and Metric Spaces

Gödel proved that the continuum hypothesis is consistent with the standard axioms of set theory . We are in a subtle area here , where there are three possibilities for a theorem : It may be provable , disprovable , or undecidable .

Author: Irving Kaplansky

Publisher: American Mathematical Society

ISBN: 9781470463847

Category: Mathematics

Page: 140

View: 677

This is a book that could profitably be read by many graduate students or by seniors in strong major programs … has a number of good features. There are many informal comments scattered between the formal development of theorems and these are done in a light and pleasant style. … There is a complete proof of the equivalence of the axiom of choice, Zorn's Lemma, and well-ordering, as well as a discussion of the use of these concepts. There is also an interesting discussion of the continuum problem … The presentation of metric spaces before topological spaces … should be welcomed by most students, since metric spaces are much closer to the ideas of Euclidean spaces with which they are already familiar. —Canadian Mathematical Bulletin Kaplansky has a well-deserved reputation for his expository talents. The selection of topics is excellent. — Lance Small, UC San Diego This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.
2020-09-10

Set Theory

Natural axioms of set theory and the continuum problem. In Proceedings of the 12th International Congress of Logic, Methodology, and Philosophy of Science, pages 43–64. King's College London Publications, 2005. J. L. Bell.

Author: Abhijit Dasgupta

Publisher: Springer Science & Business Media

ISBN: 9781461488545

Category: Mathematics

Page: 444

View: 460

What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.
2013-12-11

Philosophical Introduction to Set Theory

3 (“Theory of Deductive Systems and Theory of Multiplicities”). 61. Kunen, Set Theory: An Introduction to Independence Proofs, p. xi. 62. Cf. Kurt Gödel, The Consistency of the Continuum Hypothesis. 63. Kunen, p. 171.

Author: Stephen Pollard

Publisher: Courier Dover Publications

ISBN: 9780486805825

Category: Mathematics

Page: 192

View: 975

This unique approach maintains that set theory is the primary mechanism for ideological and theoretical unification in modern mathematics, and its technically informed discussion covers a variety of philosophical issues. 1990 edition.
2015-07-20

Notes on Set Theory

Author: Yiannis Moschovakis

Publisher: Springer Science & Business Media

ISBN: 9781475741537

Category: Mathematics

Page: 273

View: 328

What this book is about. The theory of sets is a vibrant, exciting math ematical theory, with its own basic notions, fundamental results and deep open problems, and with significant applications to other mathematical theories. At the same time, axiomatic set theory is often viewed as a foun dation ofmathematics: it is alleged that all mathematical objects are sets, and their properties can be derived from the relatively few and elegant axioms about sets. Nothing so simple-minded can be quite true, but there is little doubt that in standard, current mathematical practice, "making a notion precise" is essentially synonymous with "defining it in set theory. " Set theory is the official language of mathematics, just as mathematics is the official language of science. Like most authors of elementary, introductory books about sets, I have tried to do justice to both aspects of the subject. From straight set theory, these Notes cover the basic facts about "ab stract sets," including the Axiom of Choice, transfinite recursion, and car dinal and ordinal numbers. Somewhat less common is the inclusion of a chapter on "pointsets" which focuses on results of interest to analysts and introduces the reader to the Continuum Problem, central to set theory from the very beginning.
2013-04-17

Cantorian Set Theory and Limitation of Size

He adds : One may say that many of the results of point - set theory obtained without using the continuum hypothesis are also highly unexpected and implausible . But , true as that may be , still the situation is different there ...

Author: Michael Hallett

Publisher: Oxford University Press

ISBN: 0198532830

Category: Mathematics

Page: 343

View: 649

Cantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Frankel set theory.
1986

Combinatorial Set Theory Partition Relations for Cardinals

N. G. de Bruijn–P. Erdös [1951] A color problem for infinite graphs and a problem in the theory of relations. Akademia Amsterdam 13 (1951), 371-373. L. Bukovsky [1965) The continuum problem and powers of alephs ...

Author: P. Erdös

Publisher: Elsevier

ISBN: 0444537457

Category: Mathematics

Page: 348

View: 756

This work presents the most important combinatorial ideas in partition calculus and discusses ordinary partition relations for cardinals without the assumption of the generalized continuum hypothesis. A separate section of the book describes the main partition symbols scattered in the literature. A chapter on the applications of the combinatorial methods in partition calculus includes a section on topology with Arhangel'skii's famous result that a first countable compact Hausdorff space has cardinality, at most continuum. Several sections on set mappings are included as well as an account of recent inequalities for cardinal powers that were obtained in the wake of Silver's breakthrough result saying that the continuum hypothesis can not first fail at a singular cardinal of uncountable cofinality.
2011-08-18 By P. Erdös

Axiomatic Set Theory Part 1

certain formal statements are formally independent of the axioms of their respective theories but leave essentially untouched the question of whether ... P. Cohen, set theory and the continuum hypothesis, W. A. Benjamin, New York, 1966.

Author: Dana S. Scott

Publisher: American Mathematical Soc.

ISBN: 9780821802458

Category: Mathematics

Page: 474

View: 871

1971-12-31

From Frege to G del

On the foundations of set theory and the continuum problem JULIUS KÖNIG ( 1905a ) König's paper was written at about the taking for granted Zermelo's result that same time as Richard's ( see above , p . the set of real numbers can be ...

Author: Jean van Heijenoort

Publisher: Harvard University Press

ISBN: 0674324498

Category: Philosophy

Page: 664

View: 313