Are you saying that you are in agreement that the notion of prime is not due to a strict position on the "number line" (whatever the hell a number line is) and instead it is due to structure of the 'meta-data" you are adding when you change the appearance from "da" to "DA"?

I think that once you put connections in between the numbers (so that "3" is right between "2" and "4", where in your axiom schema right now there's nothing special connecting the three) you can form the concept of primes.

Actually I do not want to be able to have the concept of primes. That is why I left it as a counting system. Leaving it that way, I want to map the bare counting system to the Peano system's version of the counting system. After all, I am assuming that the Peano system can indeed "simulate" the bare counting system. Is this possible? Yes I think it is. Why do it? For the sake of understanding the notion of "prime" separate from the notion of the position of the "item" on the "number line."

Algorithmically speaking..... I can just as easily interpret the Peano axioms algorithmically since Peano successor function is very "algorithmic." Yes my counting system does not have position in terms that you speak of. It only has algorithmic position. Can we map this notion of algorithmic position (or step of execution - comp. sci. terms) to the "positional" stuff that we get out of the Peano axioms?

I was referring to the positional "stuff" that you get out of the Peano system.

Let try this: Just ignore positional stuff in the Peano system and try to get it to produce what we are calling a "counting system." Let us say the Peano system can do many things. One of the things it can do (we hope) is just simulate the basic counting system. we have been talking about. Now, equate these two systems Once you map them, then allow the positional stuff to come back into view on the Peano side. With it comes the notion of prime but you can not impose that notion of prime back onto the basic counting system that was mapped to the peano counting system.

Steps. How does the peano successor function produce the successor? In zero time? Does the framework allow one to talk about the successor function in terms of "steps." How does the successor function "compute" the successor of x? Is it a magical filter that pumps out numbers but does not let you look into it ?

Obviously, time is not a factor in the "ether of mathematics and abstractness" but what is preventing me from saying there are 2 steps from S(4) to S(6) ?

Actually, I think about it more and I am convinced that the Peano system definately allows you to look at what is happening in terms of algorithm. Algorithm implies steps. Perhaps Peano wanted an algorithmic viewpoint, I don't know.

Is there a way to map one axiomatic system to another? Has it been done for systems that are similar? Can it be done from one simple system to a system that is more feature rich but which still simulates the simpler system?

What exactly does it mean to map a formal axiomatic system to SOMETHING ELSE. Is this the technique employed by Godel in his most excellent work?

Still not following. What do you mean when you say you "equate the two systems"?

When you're working with the pure Peano axioms, there's noting you can say about time, space, or other complexity. If you choose a particular model of the Peano axioms, then you can talk about it.

It's done a lot, sure. A system A can be shown to be consistent relative to a (stronger) system B by constructing a model of A inside B. Kelley-Morse set theory can model ZFC (and prove its consistency too, showing that KM is actually stronger).

Which work? Godel made several major contributions to mathematics... do you mean his Incompleteness Theorem, or perhaps his earlier Completeness Theorem?

I was referring to the one which states that "for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms." (copied from wikipedia)

No. In fact those two are inequivalent -- and you should know the reason, since you just posted it: the Incompleteness Theorem. No sufficiently strong theory* can prove its own consistency, so since KM proves ZFC to be consistent the two can't be equal (unless one is inconsistent, in which case they're both equal to the theory "for all p, p" in which everything is true).

* Any theory containing Peano arithmetic is strong enough.

The Incompleteness Theorem is the more famous of the two, and it's the one you quoted. The Completeness Theorem is the one that essentially says that in first-order logic, provability <--> truth.

Godel encoding was used by Godel as follows: "Gödel used a system of Gödel numbering based on prime factorization. He first assigned a unique natural number to each basic symbol in the formal language of arithmetic he was dealing with."

Can I assign each natural number from the Peano system to the counting systems' statements which we have been talking about here? I want each step of the counting algorithm output to be assigned the associated natural number.

(sorry human language is making it difficult to be formal and to keep all my terms in proper context. Obviously, there is no existing association when I said "associated natural number" but you know what I mean...)