The foundations of mathematics may seem to be a topic curiously disconnected from the the rest of mathematics. This is not quite so. This page lists examples where questions of foundations do affect questions and problems in “real world” mathematics.
See also Wikipedia's list of statements undecidable in ZFC.
The following question is called the Whitehead problem
Every free abelian group? satisfies . Is, conversely, every abelian group that satisfies a free abelian group?
When formalizing group theory within ETCS as a set theory, then this is question is undecidable. (It is still undecidable in ZFC, equivalently in ETCS + Collection.) In some models it is true, while other models have counterexamples.