Full Idea
Informally, Peano's axioms are: 0 is a number, numbers have a successor, different numbers have different successors, 0 isn't a successor, properties of 0 which carry over to successors are properties of all numbers.
Gist of Idea
There are five Peano axioms, which can be expressed informally
Source
Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
Book Reference
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.-9
A Reaction
Each statement of the famous axioms is slightly different from the others, and I have reworded Wright to fit him in. Since the last one (the 'induction axiom') is about properties, it invites formalization in second-order logic.