Given that the Peano axiom set is stripped of the axiom that says "0 is a natural number," I still believe the underlying "form" of what was once called a "number line" would still remain completely the same despite loosing the ability to define what a number is. Peano, by stating that "1 is a natural number" has basically "encoded" a reference point into the system. However, without the axiom, a user could just define their own reference point outside of the system and just use what is left in the Peano axiom set as a "metronome." The combination of "reference" point and "metronome system" is basically enough to completely build all the numbers. In other words, from an algorithmic perspective if you have memory (for the reference point) and metronome, you can get all the numbers, addition, multiplication, "prime", etc. all in one complete magical "poof!". Honestly, I am not totally comfortable with the idea that numbers and the operations are not completely separable; however, I can intuitively understand the phenomenon.
Peano's axioms just give the user a way to input things into a recursive blackbox which then turns around and spits out a number. It is essentially an interface to recursion. It is a system which, once the recursion is kicked into gear, there is nothing you can do except wait until the "answer" comes back. You can't peer into the recursive "machinery" to glean or use "internal" information. If you take the "0 is a natural number" axiom out then the black box remains but is basically "disoriented".
I am very curious about the Peano axiomatic system minus the said axiom. How would one find practical application for this in a mathematical sense? It is a valid axiomatic system and deserves a share of study.
Furthermore, it is now my understanding that the following statements hold (more or less):
*Recursion without a reference point is basically a metronome.
*Recursion without a reference point is just unary "counting/ticking."
*Recursion can only be used to define numbers when given a seed.
*Recursion is a powerful thing (mystery) which requires an interface to be used; hence, Peano defined his axioms.
*Pure recursion does not have a reference point.
*Every time you do anything via the Peano axioms, essentially what happens is that the system starts from "0"! This is amazing!