How do you make parts of the two systems synonymous? Lets say you were going to build both systems from the ground up: A and B. Build it "modularly" as in: "build piece by piece". Start with the common pieces. How far can you get before you loose commonality?

The counting algorithm is not a formal system; it's an algorithm! As stated, I can't really make sense of your inquiry.

I can guess at other things; maybe you are interested in comparing a formalization of the theory of computation with peano arithmetic? But that doesn't seem like a meaningful comparison.

Or maybe you are interested in the fact that if I had a theory of collections, then I could use the properties of Peano arithmetic to deduce properties of the counting algorithm. Then I could attempt to reverse the process -- if I assume certain properties of the counting algorithm, then I could try and derive Peano arithmetic.

Yes: I am interested in comparing a formalization of the theory of "counting" (not the theory of computation... or are they the same?? haha don't answer that! ) with the peano arithmetic. I think it is a valid comparison.

More precisely to rephrase my thoughts and your guess:
I want to formalize "counting".. then, see what needs to be added to it this formalization to end up with something equivalent to Peano system. I believe formalized "counting" is more "basic" than Peano arithmetic. I want to explore this curiousity. If there is a "bridge" between the two systems then maybe you can do things like your second "guess" above.

Thanks for the input thusfar, it is very helpful.

Side question:
Do you think there is a "theory of counting?" or a "axiomatic counting system"? If there is no time in "axiomatic systems" then how can Peano make the successor function "stop" just long enough to give a name to the next number?

Axioms of counting based on the Peano Axioms
1. 0 is a natural number that corresponds to an empty set(peano axiom 5)
2.For every element of a non empty set A there is a corresponding natural number x that is unique, x = x. That is, equality is reflexive. (Peano axiom 1)
3. Every corresponding natural number x is an unique successor of 0 or of another corresponding natural number x. (peano axium 6)
4. For the set corresponding natural number x, there is an maximun x = n which has a successor that corresponds to no element in set A. (peano axiom 7)
5. n is the count of set A (peano axium 9)
Edit I posted an improvement on these axioms at my blog.