Successor set

A set $S\subset \mathbb{R}$ is called a successor set iff

(i) $1\in S$
(ii) $\forall n\in S$; $n+1\in S$

The set of natural numbers $\mathbb{N}$ is the smallest successor set, as for any successor set $S$, $\mathbb{N} \subset S$.

Note that $\mathbb{N}=\{1,2,3\ldots\}$ is not the only successor set. For example, the set $S=\{1,\sqrt{2},2,1+\sqrt{2},\ldots\}$ is also a successor set.

This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS