[FOM] Status of AC

[Dung Nguyen]

Dear all,


I have a question: what is the current status of AC among mathematicians and
philosophers of mathematics?

To me, it seems quite unthinkable to drop AC, since without it a large part
of mathematics would collapse (e.g. can anyone think of model theory without
Compactness or topology without Tychonoff?). I feel it is as central to
modern mathematics as infinity axiom, and much more central to Replacement
or Axiom of Foundation. Nonetheless, I heard many people support axioms like
AD, which contradicts AC. Also, in most set theory courses AC is
emphatically separated from ZF, and students are told that it remains very
controversial, while nothing is said about Axiom of Foundation!

So what's the deal with it? How controversial is it to you? And can anybody
argue against it?


Mikey Nguyen