[aithrowawaycomm]

Maybe a roundabout answer to your question, but Peano's axioms are equiconsistent with many finite set theories (even ZFC without axiom of infinity), and I do think philosophically it makes more sense to say weak axiomatic set theory + predicate calculus forms building blocks of arithmetic[1]. The idea of "number" as conceived by Frege is an equivalence class on finite sets: A ~ B <-> there is a bijection, which is in fact a good way of explaining "counting with fingers" as an especially primitive building block of arithmetic:

  {index, middle, ring} ~ 
  {apple, other apple, other other apple} ~
  {1, 2, 3}

as representatives of the class "3" etc etc, predicates would be "don't include overripe apples when you count" etc. Then additions are unions and so on, and the Peano axioms are a consequence.

[1] In my view Peano axioms are the Platonic ideal of arithmetic, after the cruft of bijections and whatnot are tossed away. I agree this is splitting hairs.