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:
“Consider what you have in the smallest chosen library. A company of the wisest and wittiest men that could be picked out of all civil countries in a thousand years have set in best order the results of their learning and wisdom. The men themselves were hid and inaccessible, solitary, impatient of interruption, fenced by etiquette; but the thought which they did not uncover in their bosom friend is here written out in transparent words to us, the strangers of another age.”
—Ralph Waldo Emerson (1803–1882)
“We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;—and that is the theory of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”
—Ralph Waldo Emerson (1803–1882)