In a sense _all_ of the axioms of a given system are undecidable. They're just given, and they're relatively arbitrarily chosen. You can create, for example a

In a sense _all_ of the axioms of a given system are undecidable. They're just given, and they're relatively arbitrarily chosen. You can create, for example a version of set theory where the starting axioms are different from ZFC (the "standard" set theory) and you can derive some of the axioms of ZFC from them. What Godel proved is that no matter what set of axioms you choose, as long as it's sufficient for expressing basic arithmetic, you can produce still produce new undecidable statements, or there will be true statements that you can't prove with it.