Full Idea
A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction.
Gist of Idea
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
A Reaction
This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'!