Discussion
The hereditarily finite sets are a subclass of the Von Neumann universe. They are a model of the axioms consisting of the axioms of set theory with the axiom of infinity replaced by its negation, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.
Notice that there are countably many hereditarily finite sets, since Vn is finite for any finite n (its cardinality is n−12, see tetration), and the union of countably many finite sets is countable.
Equivalently, a set is hereditarily finite if and only if its transitive closure is finite. Vω is also symbolized by, meaning hereditarily of cardinality less than .
Read more about this topic: Hereditarily Finite Set
Famous quotes containing the word discussion:
“There exist few things more tedious than a discussion of general ideas inflicted by author or reader upon a work of fiction.”
—Vladimir Nabokov (1899–1977)
“Power is action; the electoral principle is discussion. No political action is possible when discussion is permanently established.”
—Honoré De Balzac (1799–1850)
“We should seek by all means in our power to avoid war, by analysing possible causes, by trying to remove them, by discussion in a spirit of collaboration and good will. I cannot believe that such a programme would be rejected by the people of this country, even if it does mean the establishment of personal contact with the dictators.”
—Neville Chamberlain (1869–1940)