Hereditarily finite set
In
mathematics,
hereditarily finite sets are defined recursively as finite
sets containing hereditarily finite sets (with the
empty set as a base case). Informally, a hereditarily finite set is a finite set, the members of which are also finite sets, as are the members of those, and so on.
They are constructed by the following rules:
- {} is a hereditarily finite set
- If a1,...,ak are hereditarily finite, so is {a1,...,ak}.
The set of all hereditarily finite sets is denoted V
ω. If we denote P(S) for the
power set of S, V
ω can also be constructed by first taking the empty set written V
0, then V
1=P(V
0), V
2=P(V
1), ..., V
k=P(V
k-1)... Then
The hereditarily finite sets are a subclass of the
constructible 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.