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Successor set

From AoPSWiki

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.

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