# Infinity and the Peano axioms

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.

• Even with out infinity, it takes more than Peano to construct the reals...
– 5xum
Commented Oct 6, 2016 at 15:21
• I am not versed in axiomatics, but I don't think that introducing this extension to the reals involves any "actual infinity". You keep reasoning as before with extra numbers and a few computation rules. (By no means do you ever "reach" infinity.) The situation is similar with complex numbers, which can be seen as a mere notational convenience and do not create an imaginary universe.
– user65203
Commented Oct 6, 2016 at 15:25

$\infty$ 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.
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.