1. \(1\) is a number.
2. If \(n\) is a number, then so is \(s(n)\).
3. There is no number \(n\) such that \(s(n) = 1\).
4. Given any two numbers \(n\) and \(m\), if \(s(n) = s(m)\) then \(n = m\).
5. If a set \(X\) of numbers contains \(1\) and is closed¹ under \(s\) then \(X\) contains all numbers.

1. All you can obtain by repeated application of \(s\) will be in \(X\), there is no escape. ↩