Lev D. Beklemishev 
Provability, Computability and Reflection [PDF ebook] 

The book consists of a selection of the forms of the axiom of choice which appeared in the literature together with additional forms which were obtained in the process of writing the book. Forms which were either used often in practice, unusual, relatively unknown, or particularly weak or strong were chosen for inclusion. The book assumes a knowledge of logic and elementary set theory (von Neumann-Bemays-Godel set theory), but does include a list of definitions of set theoretical symbols and terms in the section entitled "Preliminary Definitions and Theorems".