V and Set Theory
If ω is the set of natural numbers, then Vω is the set of hereditarily finite sets, which is a model of set theory without the axiom of infinity. Vω+ω is the universe of "ordinary mathematics", and is a model of Zermelo set theory. If κ is an inaccessible cardinal, then Vκ is a model of Zermelo-Fraenkel set theory (ZFC) itself, and Vκ+1 is a model of Morse–Kelley set theory.
V is not "the set of all sets" for two reasons. First, it is not a set; although each individual stage Vα is a set, their union V is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity) demands that every set is well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (for example is Aczel's anti-foundation axiom). These non-well-founded set theories are not commonly employed, but are still possible to study.
Read more about this topic: Von Neumann Universe
Famous quotes containing the words set and/or theory:
“I never can hear a crowd of people singing and gesticulating, all together, at an Italian opera, without fancying myself at Athens, listening to that particular tragedy, by Sophocles, in which he introduces a full chorus of turkeys, who set about bewailing the death of Meleager.”
—Edgar Allan Poe (18091845)
“... liberal intellectuals ... tend to have a classical theory of politics, in which the state has a monopoly of power; hoping that those in positions of authority may prove to be enlightened men, wielding power justly, they are natural, if cautious, allies of the establishment.”
—Susan Sontag (b. 1933)