My take on this is that the axiom of choice allows you to produce infinite amounts of information, if and only if you start with infinite amounts of input.
Think about how much information is involved in presenting an infinite collection of sets! When I say even something as simple as "Let x be a real number" I'm already invoking infinitely many bits of information. Things have already gotten weird, you just haven't noticed because we've hidden it.
The axiom of choice is saying something like, okay, we somehow have this infinite pile of information sitting around; now we're allowed to interact with it. And if you don't like that, your problem might be with the infinite amount of information we started with. (Or it might not; yes-choice and no-choice are both totally valid positions.)
But in practice it doesn't matter that much because you never actually do start with infinite amounts of information, and so you don't need an infinite-amounts-of-information processor. But if we have finite amounts of information that we're pretending are infinite to simplify things, we can also process the finite amounts of information and pretend we're processing infinite information.
Think about how much information is involved in presenting an infinite collection of sets! When I say even something as simple as "Let x be a real number" I'm already invoking infinitely many bits of information. Things have already gotten weird, you just haven't noticed because we've hidden it.
The axiom of choice is saying something like, okay, we somehow have this infinite pile of information sitting around; now we're allowed to interact with it. And if you don't like that, your problem might be with the infinite amount of information we started with. (Or it might not; yes-choice and no-choice are both totally valid positions.)
But in practice it doesn't matter that much because you never actually do start with infinite amounts of information, and so you don't need an infinite-amounts-of-information processor. But if we have finite amounts of information that we're pretending are infinite to simplify things, we can also process the finite amounts of information and pretend we're processing infinite information.