Finite functions and the necessary use of large cardinals.
Finite sets and Gödel's incompleteness theorems [Book]
Finiteness and choice
We deal with weak choice principles of the form: Every "finite" family of non-empty sets has a choice function, where "finite" stands for one of several different definitions of finiteness that are not equivalent unless we assume the axiom of choice (AC). Several relations of implication and independence are established. In the process, we answer a few open questions about the relations between different definitions of finiteness.