[fxn]

Axioms are not beliefs. Axioms are a priori and you formally derive stuff from them using particular logic rules.

Generally speaking, nobody claims that the axioms of set theory are true in any ordinary sense of the word true. Does it makes sense in the real world to assert the existence of an infinite set, and have different sizes of infinities as a consequence? It is irrelevant, you assert the existence of an infinite set because it is practical from a pure mathematical point of view, because you want to model infinite sets like the natural numbers (an abstraction, do they exist? it doesn't matter to mathematicians generally speaking).

Mathematics no longer pretends to describe what is true, what holds in our reality. That idea was abandoned some time ago. A canonical example were non-euclidean geometries, that were studied in the XIXth century for the sake of it. They had applications later, but the motivation for their study and changing Euclid's axioms was formal.

[cwzwarich]

> Generally speaking, nobody claims that the axioms of set theory are true in any ordinary sense of the word true.

Mathematical platonism is a pretty widely held belief, at least amongst pure mathematicians. From Hardy's A Mathematician's Apology,

"I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply the notes of our observations."

A lot of set theorists will speculate about the truth of axioms beyond the standard axioms of ZFC, e.g. large cardinal hypotheses or projective determinacy.

[eli_gottlieb]

>Does it makes sense in the real world to assert the existence of an infinite set, and have different sizes of infinities as a consequence?

Of course it does. It's very easy for me to write down a finite-sized inductive definition of the natural numbers, which immediately gives me a finite description of all countable infinities and their size.

[Swizec]

> Does it makes sense in the real world to assert the existence of an infinite set, and have different sizes of infinities as a consequence?

Yes it does. But I might be biased because I'm a programmer and one of my jobs is adding constraints to unsolvable problems so they become solvable in the infinite set my clients actually care about.

It's surprising how easy it is for somebody to specify a problem that is unsolvable, only for it to become solvable once you ask them what they really want.