Usually inside a Complete Field CF, the natural number set in a Complete Field $(\Bbb N_{CF})$ is defined as intersection of all inductive sets in CF, where the inductive set definition is as fallow:
If our CF is (F,+,$*$,$e^+$,$e^*$,P) where P is named the positive subset of F, and K $\subset$ F, then
K is inductive in our CF if
(1) $e^+$ $\in $ K and
(2) ($\forall$x)($x\in $K $\Rightarrow$ (x + $e^*$) $\in $ K
Now suppose I define the "Naturalness" predicate in CF as
(1) K $\subset$ P
(2) $e^+$ $\in $ K and
(3) ($\forall$x)($x\in $ K $\Rightarrow$ (x + $e^*$) $\in $ K and
(4) ($\forall$n)($n\in $ K)($\forall$ x)(x $\in$ F) if (n < x and x < (n + $e^*$)) $\Rightarrow$ x $\notin $ K
My question is: Given a set S $\subset$ F then S is Naturalness $\Leftrightarrow$ S = $\Bbb N_{CF}$
Note: Thanks to @Wojowu that thanks to his comments, I made a refinement of the original question.