References for axiom of choice

Question

I would like to know the following.

• How to do mathematics, e.g. analysis, without AC
• Which theorems are provable and which are not without AC
• Proof that some propositions cannot be proved without AC
• Proof of equivalence of some propositions with AC
• What about the countable choice

Are there any good books about these?

Answer 1

Herrlich's Axiom of Choice is a good start, and to some extent, are Schechter's Handbook of Analysis and its Foundation and Fremlin's 5th volume of Measure Theory.

Consequences of the Axiom of Choice by Howard and Rubin, as is Equivalents of the Axiom of Choice by Rubin and Rubin, are good general references.

Proving things are not provable without choice requires, usually, knowledge of forcing and such. Jech covers a lot of the basic in his Set Theory and Axiom of Choice books, Halbeisen's book is also a good starting point.

Answer 2

In my naive way, I did an Amazon search for "consequences of the axiom of choice" and got these two hits:

Consequences of the Axiom of Choice (Mathematical Surveys & Monographs) by Paul Howard | Jun 30, 1998 (\$80).

Zermelo's Axiom of Choice: Its Origins, Development, and Influence (Dover Books on Mathematics) by Gregory H. Moore | Feb 21, 2013 (less than \$20 paperbaack).