Reference request, self study of cardinals and cardinal arithmetic without AC I'm looking for references (books/lecture notes) for :


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*Cardinality without choice, Scott's trick;

*Cardinal arithmetic without choice.


Any suggestions? Thanks in advance.
 A: *

*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:


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*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:


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*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. 

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