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 Succesor Set because 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.

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Successor set