This is a hard one, i need . (this is dealing in philosophy, but i know involves math as well.)

what does someone mean when they say they accept the axiom of choice in the zermelo-frankel framework?

if you could maybe explain the axiom of choice in the zermelo-frankel framework?

This is a hard one. The axiom basically claims that for each set of things, there is a way to select one element from it. When the elements are numbered (ordinals) it's really easy, you could for instance select the smallest one. When they are not numbers, it is not obvious that you can choose in a non random way (randomly choosing doesn't count I guess). However, the axiom states that there shall be a way to do it.

The way to do it is called the choice function f. So if you set of things is called A, then f(A) is part of A (of course) and also not empty.

If you have many sets A, called X (a set of sets) then f(A) is part of A for every set A in X.

The wikipedia article is useful. Especially the comparison of shoes vs. socks. With shoes you can easily choose e.g., the left one, but with socks, you can't...