Infinity and the Peano axioms

Question

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.

Answer 1

To construct the reals starting with Peano's Axioms, you will need more axioms of set theory than just Choice. You will also need to at least be able to construct Cartesian products, power sets, subsets and functions. And while the $\infty$ symbol may be a handy shorthand, strictly speaking, it isn't necessary in formal proofs.

Answer 2

$\infty$ is not a real number, so it's not surprising that it violates axioms for the real numbers. But statements about the extended reals are actually statements about the reals, to which the axioms do apply.