Set Theory and the Continuum Hypothesis

by Paul Cohen

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. The independence of the continuum hypothesis is the focus of this study by Paul J. Cohen. It presents not only an accessible technical explanation of the author's landmark proof but also a fine introduction to mathematical logic. An emeritus professor of mathematics at Stanford University, Dr. Cohen won two of the most prestigious awards in in 1964, he was awarded the American Mathematical Society's Bôcher Prize for analysis; and in 1966, he received the Fields Medal for Logic.
In this volume, the distinguished mathematician offers an exposition of set theory and the continuum hypothesis that employs intuitive explanations as well as detailed proofs. The self-contained treatment includes background material in logic and axiomatic set theory as well as an account of Kurt Gödel's proof of the consistency of the continuum hypothesis. An invaluable reference book for mathematicians and mathematical theorists, this text is suitable for graduate and postgraduate students and is rich with hints and ideas that will lead readers to further work in mathematical logic.

Genres: Mathematics, Philosophy, Logic, Nonfiction, Science, Stem

190 pages, Paperback

First published August 1, 1966

About the author

Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.

Reviews

[Thor Nordahl · Rating 4 out of 5]

Tok meg 5 måneder med lockdown å komme meg igjennom denne greia...

Hva har jeg lært? All matematikk vi kjenner til kan modelleres på en måte som gjør helt usaklige påstander sanne, men det kan også modelleres på en måte som gjør helt fullstendig saklige påstander usanne. I tillegg kan man konstruere matematisk teori på en måte som kan gjøre ting usanne og sanne avhengig av hva du baserer konstruksjonen på, som omtrent er det samme som å si "vet da faen jeg".

Var det verdt det? Umulig å svare på (hehe).

Får èn stjerne trekk for èn typo som fikk meg til å stille spørsmål ved forstanden min i 3 dager i strekk.

[J. · Rating 5 out of 5]

Now this is how to really understand forcing. No offense to Badiou, obviously, for not making this foundational book dispensable. Even (& especially) careful readers of Being and Event should rather follow his example and work through this. Confirms Cohen's vaunted pedagogical elucidatory talents with compound interest. Vastly easier than Cohen's original journal papers on forcing, you'll be able to follow this if you're as sharp as a Harvard undergrad (and who isn't?)

[Nick Black]

2009-02-13. Thanks, Dover, for reprinting this classic (a perennial favorite on http://outofprintmath.blogspot.com)!

[meadow · Rating 4 out of 5]

well i'm going to read this book again in the near future to get a more complete grasp on all the topics. i read it too slowly! school got in the way. still, though its only 150 pages this book contains an astounding amount of, uh, good theorems and shit. would like if it was typeset using LaTeX, but hey i'm just happy dover is publishing it again. (and for so cheap!). a few typos. a few of the proofs could have maybe used a bit more detail. all in all though, really well organized, compact, and cohen did a very good job presenting the intuition behind the proofs.

[Jared Davis]

Following a presumptuous page and a half introduction, Cohen presents the standard symbols of first order logic - "and" "or" "not" "if...then" "if and only if" - and then the two second order quantifiers - "some" "all" - bringing the reader to confront Godel's completeness theorem ( roughly, the cardinality of a model does not exceed the cardinality of the statement set it interprets ) on page 11.

I'm saved by having studied logic and the completeness theorems before.... and by Cohen's adept skill at exposing the core techniques bolstering, well, basically all of mathematics.

[Richard Marney · Rating 5 out of 5]

A wonderful, yet humbling adventure to explore the infinite! Requires careful consideration and additional effort to digest fully the theorems but the work is worthwhile. Who says mathematics is not beautiful!!😊

[Harry Vincent · Rating 5 out of 5]

9/10 - brilliant intro to model theory in context. I will return to fully understand forcing some other time.