Reference request, self study of cardinals and cardinal arithmetic without AC

Question

I'm looking for references (books/lecture notes) for :

Cardinality without choice, Scott's trick;
Cardinal arithmetic without choice.

Any suggestions? Thanks in advance.

Asaf Karagila · Accepted Answer · 2013-06-01T17:37:30.773

6

Jech, The Axiom of Choice.
Herrlich, The Axiom of Choice.
Halbeisen, Combinatorial Set Theory.
Jech, Set Theory, 3rd Millennium Edition.

Jech's (first) book is kinda old, and some progress has been made since then, but I don't think there has been a lot that we can say about cardinal arithmetic that was discovered since that book was published (on their order, other structure properties and complexities - sure).

Herrlich's book is not a set theoretical book per se, but it has a reasonable chapter about basic failures of cardinal arithmetics. In particular with the existence of infinite Dedekind-finite sets, which give us a great source of interest for counterexamples.

For the most part, let me tell you what we know about cardinal arithmetic without the axiom of choice:

The basic addition, multiplication and exponentiation is well-defined as finitary operations. Those are easily found in any set theoretical textbook.
Everything else can fail miserably.

Some interesting papers:

Rubin, Jean E. Non-constructive properties of cardinal numbers. Israel J. Math. 10 (1971), 504–525.
Halbeisen, Lorenz; Shelah, Saharon Consequences of arithmetic for set theory. J. Symbolic Logic 59 (1994), no. 1, 30–40.
Halbeisen, Lorenz; Shelah, Saharon Relations between some cardinals in the absence of the axiom of choice. Bull. Symbolic Logic 7 (2001), no. 2, 237–261.

edited Jun 01 '13 at 17:37

answered Jun 01 '13 at 16:53

Asaf Karagila

393,674

3

You should also look at Lorenz Halbeisen's "Combinatorial set theory", and the references within. The book is downloadable from the author's page, here. In particular, chapter 4 contains several hard to find or new results. – Andrés E. Caicedo Jun 01 '13 at 17:20
@Andres: I had a hunch that I'm forgetting something! – Asaf Karagila Jun 01 '13 at 17:29
Incidentally, I ran into Rubin's paper just last night. – Asaf Karagila Jun 01 '13 at 17:37
1

I would recommend to begin with Halbeisen's book rather than the Halbeisen-Shelah papers. The exposition in the book is cleaner, and may be easier on a first pass. – Andrés E. Caicedo Jun 01 '13 at 17:42
@Andres, I agree. I recommend start with all the books before proceeding to the papers. – Asaf Karagila Jun 01 '13 at 17:45
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Many thanks. I had a quick look to the books and they are exactly what I was looking for. – M.G Jun 01 '13 at 17:52

score 3 · Answer 2 · answered Jun 01 '13 at 16:43

3

Azriel Lévy's "Basic Set Theory" discusses Scott's trick, and does some discussion of choiceless arithmetic.

answered Jun 01 '13 at 16:43

Cameron Buie

102,994

Reference request, self study of cardinals and cardinal arithmetic without AC

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