Following reply #4, here is adaption of James Henle's work in his

__An Outline of Set Theory__.

- $\displaystyle \neg \left( {\exists n} \right)\left[ {n < 0} \right]$
- $\displaystyle \left( {\forall n} \right)\left[ {n < S(n)} \right] \wedge \neg \left( {\exists m} \right)\left[ {n < m < S(n)} \right]$
- $\displaystyle 1 = S(0)$

Note there is a reversal in the order of statements.

BTW: $\displaystyle S(n)$ stands for

*successor of n*.

Henle defines $\displaystyle S(n)$ as:

For any

**set** $\displaystyle n,~$ $\displaystyle S(n)=n\cup\{n\}$