Reference request, self study of cardinals and cardinal arithmetic without AC
- 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.
Azriel Lévy's "Basic Set Theory" discusses Scott's trick, and does some discussion of choiceless arithmetic.