For sure - as I understand it in most formulations you throw out lots of sets but keep all the finite sets of natural numbers, the set of even numbers, the set of prime numbers etc
One thing I never found satisfactory is that any axiom system like this has to define things in terms of decidability etc, so it's "more verbose" (or less axiomlike) than ZFC
One thing I never found satisfactory is that any axiom system like this has to define things in terms of decidability etc, so it's "more verbose" (or less axiomlike) than ZFC