According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but realized that he needed things such as an infinity axiom to get his reduction off the ground. But of course, that there are infinite collections is not just a matter of logic. So it seems that Russell just had to stipulate an infinity axiom. (This is just background.)
So, in modern set theory, is the axiom of infinity just stipulated? Or is there an argument for its truth?
Some directions:
G. Boolos derived the ZFC axioms from the iterative conception of set, and thus gave a motivation or argument in favor of the axiom of infinity.
Or, someone might think, as Cantor did, that all consistent mathematical results have (material?) instantiations in nature. Much of mathematics is dependent on the natural numbers, the real numbers, etc., and thus there is reason to accept axioms of infinity.
There are some similar threads to mine:
Best Answer
If you believe in the set of natural numbers you already accept the axiom of infinity. For most mathematician existence of the set of natural numbers is an intuitively clear fact that doesn't need an argument so mathematicians are typically not bothered with the axiom. In addition lots of classical mathematics depends on such infinite concepts.
The question of accepting or rejecting such an axiom is mainly interesting for philosophers not mathematicians. One can reject the axiom of infinity (such people are often called finitists) but most mathematicians do not. They believe in the existence of the set of natural numbers and therefore see the axiom of infinity as a trivially true fact.