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

  1. Jech, The Axiom of Choice.
  2. Herrlich, The Axiom of Choice.
  3. Halbeisen, Combinatorial Set Theory.
  4. 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:

  1. Rubin, Jean E. Non-constructive properties of cardinal numbers. Israel J. Math. 10 (1971), 504–525.
  2. Halbeisen, Lorenz; Shelah, Saharon Consequences of arithmetic for set theory. J. Symbolic Logic 59 (1994), no. 1, 30–40.
  3. 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.