What are some examples of successor sets besides the set of

What are some examples of successor sets besides the set of natural numbers.

Solution

By the Axiom of Infinity, there is at least one successor set; call it S. A successor subset

of S is, by definition, a subset of S that is a successor set. Let f be the collection of all successor subsets of S; i.e., f = {X : X .S & suc[X]}.

Claim: f is the smallest successor set.

Proving this divides into showing the following two:

(1) f is a successor set;

(2) f is included in every successor set.

Proof of (1): f is a successor set, since it is the intersection of a collection of successor sets . In order to show (2), we first show the following:

Claim: f is the smallest element of f, which is to say it is the smallest successor subset of S.

Proof: we already showed that f is a successor set. It is also a subset of

S, since it is the intersection of a collection of subsets of S . So

f is itself a successor subset of S. Also, f is included in every

element of f . Thus, f is a successor subset of S, and is

included in every successor subset of S. So, by definition, f is the

smallest successor subset of S.