Definition
The cumulative hierarchy is a collection of sets Vα indexed by the class of ordinal numbers, in particular, Vα is the set of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α; Vα may be defined by transfinite recursion as follows:
- Let V0 be the empty set, {}:
- For any ordinal number β, let Vβ+1 be the power set of Vβ:
- For any limit ordinal λ, let Vλ be the union of all the V-stages so far:
A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines "the set x is in Vα".
The class V is defined to be the union of all the V-stages:
An equivalent definition sets
for each ordinal α, where is the powerset of .
The rank of a set S is the smallest α such that
Read more about this topic: Von Neumann Universe
Famous quotes containing the word definition:
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)