I was recently reading a post basically discussing an "intuition" that Goldbach's Conjecture may be a statement which is undecidable (the post does not specify which axiomatic system the statement is undecidable in, I'm assuming Peano Arithmetic). I read the relevant passage in the article mentioned by the post, and as a side note, I believe the "intuition" given in said article is quite...hand-wavy. I don't see how this is "intuitive." Anyway, to quote wikipedia:
"The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment."
My question is this: Suppose we have a non-standard model of PA, $M$, with initial segment $m \cong \mathbb{N}$ via $f:m \rightarrow \mathbb{N}$. Clearly we may define $x \in m$ to be "prime" if and only if $f(x)$ is prime in the usual sense. But what about $y \in M-m$? In general, how would one go about defining a "prime number" in a non-standard model?
This question is probably very dependent on specific non-standard models. The reason I ask is this: Suppose one were to attempt to prove independence of GC with PA via finding models $M, Q$ in which GC is true and false respectively. If one were unable to define primes outside the initial segments of these models $m,q$, essentially you'd be stuck with the original conjecture in the standard model due to the existence of isomorphisms $f:m \rightarrow \mathbb{N}$ and $g: q \rightarrow \mathbb{N}$, so you might as well attack the conjecture in the standard sense. Alternatively, if one could find such a definition, it may be easier to find a counterexample outside the initial segments (or it may not, this is speculative, one could at least try).
You still have addition and multiplication in your non-standard model. Hence you may define $$ \operatorname{prime}(p)\stackrel{\text{def}}\iff p>1 \land \forall a\forall b(ab=p\to (a=1\lor b=1))$$ (even though that actually defines irreducible instead of prime)