Successor set

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

(i) $1\in S$

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

Note that the set of natural numbers $\mathbb{N}=\{1,2,3\ldots\}$ is not the only successor set.

For example, the set $S=\{1,\sqrt{2},2,1+\sqrt{2},\ldots\}$ is also a successor set. The set $\mathbb{N}$ is called the Smallest Succesor Set because for any $\mathbb{F}$ that is a succesor set $\mathbb{N} \subset \mathbb{F}$

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Successor set