Full Idea
The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence.
Gist of Idea
Natural numbers have zero, unique successors, unending, no circling back, and no strays
Source
Peter Smith (Intro to Gödel's Theorems [2007], 01.1)
Book Reference
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.1
A Reaction
These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems.