Quote by CRGreathouse
I think you just did that  mathematicians would use the term "axiom schema", that is, each number is its own axiom:
Axiom 1: 1 is a natural number.
Axiom 2: 2 is a natural number.
. . .
OK, so now you have a system where you cannot add or take the successor, but you have the natural numbers. ....

Finally I had one last important thought. Given the system described where you have the natural numbers but you can not add or take the successor, we should be able to map the system to the system where by you can build the natural numbers. If such a mapping is "formalized" then the problem appears. On the one hand you have a system where "prime" is not defined and on the other hand you have a system where "prime" can be defined. They are mapped to each other and so now is there a paradox?
I am thinking about the utility of mapping similar to what is used by Godel in his famous proof.
Again for clarity: we defined a "counting" style, infinite statement axiomatic system which you have no notion of multiplication nor successor function (as in the above posts). We have another system like Peano. Both systems produce something that lies on the same place on the number line. We use mapping to link the two systems through the "number line." Now, despite the mapping (if it is possible), you can not impose the notion of prime on the simpler system. Hence, the notion of "prime" is directly related to the mechanisms of addition/multiplication or other operations.... NOT the actually position on the number line thing.