In my measure theory class my professor told us that all of our lessons would require at most the Peano axioms and the axiom of choice. But, our professor recently introduced the extended reals where we are allowed to do operations with the infinity symbol. This appears to create a problem.
First, I don't see how the existence of an actual infinity is implied by any combination of the Peano axioms. Second, I think that arithmetic with an actual infinity violates the Peano axioms.
It's not clear to me how my second point would not be obvious.