>>12403646aleph-null is the placeholder for the cardinality of the naturals. An interesting hypothesis (the continuum hypothesis, or CH) is that the cardinality of the powerset of naturals is aleph-one, the next infinity. Some work by Godel and Cohen has shown that CH is independent of ZFC (the standard set theory)
The set of all naturals is given axiomatically by the axiom of infinity. And there are plenty of set theories other than the "standard" ZFC, and even zermelo-fraenkel-esque set theories have variants that don't have the axiom of infinity, or have an axiom which is the negation of the axiom of infinity.
As a consequence of some zermelo-fraenkel axioms, you are guaranteed the set of naturals, and you are guaranteed any set that is mapped to by a function on the naturals.
The axiom of infinity essentially allows us to say
-0 is in the set of naturals
-the successor of a natural is a natural, and so is in the set of naturals
Since there is no natural without a successor, we have an infinite set.
But if you prefer finitism you can look into those axiomatic systems.
I personally am not that interested in them at the moment. But there are people on 4chan who like to argue and will tell you that you can't have infinite sets, and there are legitimate mathematicians and philosophers who do work on the non-infinite branch of set theory
