It is interesting that the relation "on the same successor (adding 1 repeatedly) number line" isn't expressible in first-order predicate calculus (the type of logic that Godel's thm. is talkign about).

It is also interesting that there is at least one model of that first-order logic+Peano axioms that has infinitely many disconnected successor lines - http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem (Lowenheim-Skolem) . That is, starting with 0, adding 1 repeatedly, you can never reach most of the numbers.

But what the story seems to posit is merely a finitely large number on the same number line as 0, whose successor isn't on that line. But the line is defined (in a statement too powerful for 1st order logic, by me) as all the things reachable from 0 by repeatedly adding one, so that's impossible.

Or, looked at another way, you allude to an algorithm for adding 1 to what is merely a very long string of digits. Ripple carry counting, for example, will always take digit strings to digit strings. An algorithm is likely telling you much more detail than Peano+1st-order does.