As long as the mathematics you do within your chosen set of axioms is internally consistent you are doing “real” mathematics. And different choices of axioms a

As long as the mathematics you do within your chosen set of axioms is internally consistent you are doing “real” mathematics. And different choices of axioms and logics will give you different flavours of mathematics. That may or may not be beautiful or elegant or useful according to your specific tastes and needs. It just happens to be the case that the Axiom of Infinity has been a popular choice among mathematicians. While computer scientists working within type theory and proof theory usually prefer to use mathematics based on a different axiomatic foundations than what a typical career mathematician would choose to be successful in getting jobs/grants etc. However it is all “real” mathematics. Just different branches off the same tree.