>the real world does not contain infinities
I have no idea what "real world" even means, but I'd say that infinity does not exist within anything that can be conceptualized.
Yet math is consistently being conceptualized?
Enabling clear thinking about concepts outside of the real world is the while point of math.
What does this mean?
Sure, how else could we capture it or convey it to others? The thing is that I don't see it as that different than conceptualizing things in the "real world". In the latter, conceptualization is done on input of the senses, while in math conceptualization is done on inputs from a higher level of abstraction. In both cases, something is defined and then manipulated for something useful. It's just that I believe that whatever is defined automatically becomes a finite concept otherwise it could not be grasped and manipulated, and a finite infinity seems problematic...
I don't think there will ever be a situation like: "According to the axiom of choice, this photon should go to the left, but according to the axiom of something else, this photon should go to the right." Or: "If there is such a cardinal X that mumbo-jumbo(X) is true, then the photon should go to the left, otherwise it should go to the right."
Therefore, this part of math is just playing with words. An interesting game, especially if you can get paid for it, and it can provide indirect benefits such as greater insight into how to correctly play with words, but ultimately just a game with no correspondence to the "real world".