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I would like to request a book recommendation for axiomatic set theory that can be read after Kleene's "Introduction to Metamathematics". I like the style of "Introduction to Metamathematics" so similar style would be preferred. I would like this book to be as rigorous as possible. Due to the fact that I would like to work in the field of axiomatizing physical theories, I would like this book to contain not so many advanced topics but more information about different axiomatizations of set theory, considering their advantages, disadvantages, differences, and intuitions behind them. Of course, it may well be possible that such a book does not exist but then maybe it is possible to gain such knowledge from several books, which is perfectly fine.

I would appreciate any pieces of advice and suggestions.

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I know of one book that I think does this admirably, and that is is W.V. Quine's Set Theory and its Logic. The book develops its main concepts through a pretty minimal set theory (barely more than general set theory as this article calls it), adding new axioms when the mathematics being described needs it. Part III takes a very nice survey of various other set theories, including NBG, a form of Ackermann set theory, typed set theory, and Quine's own NF and ML theories.

There are some major caveats, as much as I like this book.

  1. The notation is very strange; Quine remained fond, for a very long time, of the kind of notation used by Russell and Whitehead, and while he made a few modern notational concessions in this book, it's weird getting used to entirely new ways of writing familiar things. This may not be a big deal if you've just been reading Kleene, though, and there is an index of theorems and notations in the back.

  2. Quine uses some odd dummy terms as an expositional device---$y\in\{x:\varphi(x)\}$ is just a fancy way of writing $\varphi(y)$---which leads to some succinct ways of stating things, but means that one needs to exercise caution in distinguishing actual terms from term-like abbreviations. Once you get the hang of it, though, you can communicate a lot about a theory's comprehension scheme in a very succinct way.

  3. You're not going to learn a huge amount about ordinal and cardinal numbers. This isn't surprising since Quine is trying not to commit to much in the way of axioms, but it means that if you want to go on to read other literature that does make more use of such theorems then this book will not prepare you well.

In spite of the long list of warnings I think it's a rewarding read, but you might want to flip through it first to make sure it's a good fit.

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The place where your choice of set theoretic axioms make the most difference to "applied math" is in real analysis. Do there exist non-measureable sets, etc?

A book on this is "Set Theoretical Aspects of Real Analysis" by Alexander B. Kharazishvili.

A nice overview of modern logic and set theory, with just enough technical detail to make it clear but not too much to be overwhelming, is "A tour through mathematical logic" by Robert S. Wolf.

From a more philosophical point of view, the articles "the set-theoretic multiverse" by Joel D. Hamkins and "Believing the axioms" by Penelope Maddy is worth reading.

Finally, if you haven't yet read about nonstandard analysis that could be worth it. A great place to start is this blog post by Terrence Tao.

As for applications to physics, to quote Richard Hamming: "Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane." Who knows what he would have said about different axioms of set theory...

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