I think you are begging the question by accepting ZF (and that basis for algebra)and then rejecting any possible alternatives by saying you'd lose algebra. This

I think you are begging the question by accepting ZF (and that basis for algebra)and then rejecting any possible alternatives by saying you'd lose algebra. This is not true, even the article mentioned that much of math can be founded on infinity as potential, just a different approach. The issue really is a question of philosophy and what are axioms, what is knowledge and what is math.