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.

Another one is Presburger Arithmetic, which is Peano Arithmetic minus the multiplication. What makes it interesting (and useful) is that this removal makes the theory decidable.

https://en.wikipedia.org/wiki/Presburger_arithmetic

Skolem arithmetic is decidable too: https://en.wikipedia.org/wiki/Skolem_arithmetic

I'm wondering whether there are decidable first-order theories about the natural numbers that are stronger than either Skolem or Presburger arithmetic, that presumably use more powerful number theory. Ask "Deep Research"?

[edit] Found something without AI help: The theory of real-closed fields is decidable, PLUS the theory of p-adically closed fields is also decidable - then combined with Hasse's Principle, this might take you beyond Skolem.

[edit] Speculating about something else: Is there a decidable first-order theory of some aspects of analytic number theory, like Dirichlet series? That might also take you beyond Skolem. https://en.wikipedia.org/wiki/Analytic_number_theory#Methods...

I recently learned about https://en.wikipedia.org/wiki/Self-verifying_theories which gives you most of multiplication while still being decidable, which is pretty crazy.

That's cool, but where does it say it's decidable?

Not on that Wikipedia page, but you might want to have a look at the papers?

Some decidable extensions of Skolem and Presburger, searched for and found by ChatGPT: https://chatgpt.com/share/67e1d302-c930-800f-bc2a-85bdc60563...

There are no specific extensions mentioned, a bunch of math symbol rendering issues, and what seems like maybe some hallucinations? Thanks for proving once again how useless chatgpt is if you're not already an expert on what you're asking it

Conway's Surreal Numbers https://en.wikipedia.org/wiki/Surreal_number#Arithmetic

There are many axiom systems for natural numbers. My favorites are 1. Heyting Arithmetic, and 2. In category theory, one can characterise the natural numbers and the initial algebra of the X |–> X+1. functor.

Sure, one can, but does it help with anything? Just curious.

It's up to you if you think it's "better" but it's answering the question of whether Peano axioms are the only fundamental structure

Category Theory itself assumes the existence of natural numbers. My point was about whether their futher "axiomatization" via a functor makes it easier to prove theorems or make discoveries.