Comment by IsTom

It's the same way you don't need to have GCD in stdlib to say that you can compute GCD in C++. You can make your own using parts given.

You don't need to add any axioms, you just build some sets to represent numbers and make operations that act the same way as arithmetic, define some equality relations. Then you derive rules of arithmetic for your handcrafted arithmetic using ZF axioms and you're good. You get axioms of arithmetic derived from your regular axioms without adding them as new axioms to your theory.