Thursday, 1 September 2011

New book: Set Theory, Arithmetic, and Foundations of Mathematics

Yesterday the announcement of a new book was sent around at the FOM list, and as it looks like a very interesting book, I thought I'd put a notice of it here at M-Phi too. It is edited by Juliette Kennedy and Roman Kossak, and the (somewhat vague) title is Set Theory, Arithmetic and Foundations of Mathematics. Here is the table of contents:

1. Introduction - Juliette Kennedy and Roman Kossak;
2. Historical remarks on Suslin's problem - Akihiro Kanamori;
3. The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture - W. Hugh Woodin;
4. ω-Models of finite set theory - Ali Enayat, James H. Schmerl and Albert Visser;
5. Tennenbaum's theorem for models of arithmetic - Richard Kaye
6. Hierarchies of subsystems of weak arithmetic - Shahram Mohsenipour;
7. Diophantine correct open induction - Sidney Raffer;
8. Tennenbaum's theorem and recursive reducts - James H. Schmerl;
9. History of constructivism in the 20th century - A. S. Troelstra;
10. A very short history of ultrafinitism - Rose M. Cherubin and Mirco A. Mannucci;
11. Sue Toledo's notes of her conversations with Gödel in 1972–1975 - Sue Toledo;
12. Stanley Tennenbaum's Socrates - Curtis Franks;
13. Tennenbaum's proof of the irrationality of √2.

I'm not sure what the idea is behind grouping this particular collection of papers (I have not had the chance to check it out, there's probably something on this at the introduction), but it does look like many of these papers are a must-read. I'm particularly interested in the papers concerning non-standard models of arithmetic and Tennenbaum's theorem (full disclosure: Juliette Kennedy and I had a very interesting correspondence on the topic a few years ago), but the set-theory section is also high-power stuff for sure!


  1. Catarina, thanks! I saw this on FOM, but you beat me to it.
    (There were some html tags putting background colour behind the text, so chopped them out. That ok?)

  2. Hi Jeff, that's very ok! This had happened to me before, and at that time I had had the patience to remove the color tags; this time I was writing the post in the train, and said to myself "I'll do it later" (it's so tedious!).

    But anyway, this looks like an interesting book indeed. As I said yesterday on G+ in reply to somebody who asked me what I thought of the idea of 'foundations of mathematics': I think the general project is at its best when it explores its own limitations, such as the phenomena of incompleteness and non-categoricity. Now, it looks like there is plenty of both in this volume, hence my interest.

    I'll probably write a blog post soon on how incompleteness and non-categoricity are the duals of each other (it's a section in my book, I'll probably get there next week at some point).

  3. The book looks very interesting, definitely. Thanks for highlighting it.