Set Theory and Logic (Dover Books on Mathematics)

This is an ideal painless introduction to standard logic and set theory for anyone with a couple of years of undergraduate pure mathematics background. This 1963 book by Robert Roth Stoll is more than twice as big as the author's 1961 "Sets, Logic and Axiomatic Theories", which it is an expansion of. The 1961 book was already very good, but this greatly expanded 1963 edition is much more comprehensive, and still very beginner-friendly, not one of those macho books which inflict the maximum pain on readers. Before proceeding to the more painful advanced logic and set theory books, it's probably a really good idea to first get relaxed and comfortable with this friendly introduction for a while.Instead of presenting formal logic first, as most of the advanced books do, Stoll starts with topics which are more familiar from general mathematics. Chapter 1 covers sets, relations, functions and order in a very elementary way. Chapter 2 (less elementary) presents natural numbers, cardinal numbers, ordinal numbers, and the axiom of choice. Chapter 3 presents rational and real numbers. The real numbers are constructed here using Cauchy sequences rather than Dedekind cuts.Then the logic starts in Chapter 4 with basic propositional and predicate calculus. Chapters 5, 6, 7 and 8 are concerned with the axiomatization of set theory and algebra. Finally Chapter 9 presents first-order languages and metamathematics, but not in too much depth. This lightweight introduction helps the reader to be comfortable with these concepts before optionally progressing to the more heavyweight model theory books.One consequence of presenting elementary set theorems before presenting set theory axioms is that some of Stoll's theorem's precede the axioms which they are based on. For example, theorems 4.4 and 4.5 on pages 91-93 use the countable axiom of choice, which is not introduced until pages 111-118 informally, and on page 302 formally. Interestingly, Stoll states on page 403 that David Hilbert believed that a metatheory "should belong to intuitive and informal mathematics". Stoll then states the following about Hilbert's view."Further, its theorems [....] must be understood and the deductions must carry conviction. To help ensure the latter, all controversial principles of reasoning such as the axiom of choice must not be used."So this is a bit of meta-metamathematics by Hilbert. You don't read in many books about Hilbert's suspicions about the axiom of choice.One of the best things about this book is the presentation of a natural deduction logical calculus on page 183 in the tabular style of Suppes and Lemmon. This is, in my opinion, by far the best way to do propositional and predicate calculus rigorously and with minimum effort.There is some model theory and semantics in this book, but not too much. The set theory is standard ZF plus the axiom of choice. Not too much is said about interpretations, but just enough to prepare the reader for later studying model theory if they are that way inclined.

Nice book with a clear language and a bunch of examples and exercises in order for you to practice

Very good coverage of set theory and logic.It's very readable, and also covers basic real analysis and the foundations of arithmetic.Ever wonder why a negative times a negative is a positive?i.e. why (-1)*(-1) === 1, this book answers this.

This is a good intro but if you're not a trained mathematician be prepared to do additional reading to understand it, and sometimes Stoll is not so clear about what that reading is, so you may need to dig around. Found myself finally digging into Gauss to get through the 1st chapter so it's a healthy pain and Bob Stoll's writing is clear, moreover there are exercises which most works on axiomatic set theory lack. Came to Stoll via a reference in E.F. Codd, glad I did.

Good introduction, especially the first few chapters. It gets increasingly technical, as you might expect. Some of the later chapters have more OCR errors than I would like.

I essentially agree with what has been said. This book might be too watered down for hard core mathematicians but it's perfect for philosophers and others with an interest in math who lack a strong math background. I study philosophy and logic, and hence have some prior background but not enough to make me competent in set theory. This book is really clear. Several key concepts are elaborated and there are plenty of proofs.

My son and I have been working through this. It is one of those rare-advanced math books that with a little effort and grit can work through and begin to appreciate Cantor's and Godel's work.

This is a good book for an introductory course on the subject. It contains the basic topics, but if you are looking for strong introduction to mathematical logic, maybe this is not a good choice. In any case, you can take it for granted since contains a very good selection of topics and perhaps you can support it with other books.