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Single Idea 10096

[from 'Philosophies of Mathematics' by A.George / D.J.Velleman, in 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets ]

Full Idea

ZFC is a theory concerned only with sets. Even the elements of all of the sets studied in ZFC are also sets (whose elements are also sets, and so on). This rests on one clearly pure set, the empty set Φ. ..Mathematics only needs pure sets.

Clarification

ZFC abbreviates 'Zermelo-Fraenkel with Choice'

Gist of Idea

Even the elements of sets in ZFC are sets, resting on the pure empty set

Source

A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)

Book Reference

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.48


A Reaction

This makes ZFC a much more metaphysically comfortable way to think about sets, because it can be viewed entirely formally. It is rather hard to disentangle a chair from the singleton set of that chair.