The infinitist schizos who think ZFC is enough to characterize what a set is and enough to base whole mathematics on must not have heard of compactness theorem in first order logic.
Provided ZFC is consistent (an already extremely dubious assumption), the universe of sets which satisfies ZFC axioms may have
1. Infinite natural numbers. I.e. natural numbers that are larger than 1,2,3,4,5,6....
2. Infinitesimal rational numbers (i.e. rational numbers x such that 0<x<1/n for all n (INCLUDING n infinitely large)).
3. The set {{{....}}}} (i.e. a set whose only element is a set whose only element is a set and so on ad infinitum). Contrary to popular belief, the axiom of regularity does not prohibit such sets.
4. Countably many real numbers
These schizos will look you straight into the eyes and tell you that they don't need to tell you what a set is, and that ZFC is enough.
Provided ZFC is consistent (an already extremely dubious assumption), the universe of sets which satisfies ZFC axioms may have
1. Infinite natural numbers. I.e. natural numbers that are larger than 1,2,3,4,5,6....
2. Infinitesimal rational numbers (i.e. rational numbers x such that 0<x<1/n for all n (INCLUDING n infinitely large)).
3. The set {{{....}}}} (i.e. a set whose only element is a set whose only element is a set and so on ad infinitum). Contrary to popular belief, the axiom of regularity does not prohibit such sets.
4. Countably many real numbers
These schizos will look you straight into the eyes and tell you that they don't need to tell you what a set is, and that ZFC is enough.
