Truth of Goldbach conjecture and its relationship with provability from Peano axioms. In lecture notes from Lou van den Dries, page 3 here - http://www.math.uiuc.edu/~vddries/main.pdf, it is said:
"However, in our present state of knowledge, $GC$ (Goldbach conjecture) might be true in $(\mathbb{N}; 0, 1, +, ·, <)$, but not provable from those axioms (Peano axioms). (On the other hand, once you know what exactly we mean by provable from the Peano axioms, you will see that if $GC$ is provable from those axioms, then $GC$ is true in $(\mathbb{N}; 0, 1, +, ·, <)$, and that if GC is false in $(\mathbb{N}; 0, 1, +, ·, <)$, then its negation $\neg GC$ is provable from those axioms.)"
Could someone please explain me why this is true? I don't know where to start. I'm sure if I go through the whole notes I might get a better idea of why this is true, but given my understanding that's quite far away from where I am now, and this is a nice concrete example of something interesting, that I'd like to understand now. A pointer to where exactly in the notes I should look would be appreciated as well.
 A: The key fact is:

Every model of $PA$ is an end extension of $\mathbb{N}$.

This means that any statement using only bounded quantifiers which is true in $\mathbb{N}$ is true in every model of $PA$.
The other key fact is:

Every element of $\mathbb{N}$ is definable in every model of $PA$, by a bounded formula.

For instance, $0$ is defined as the unique $x$ such that "$\forall y<x(y\not=y)$". Note that this does not mean that the set $\mathbb{N}$ is definable in $M$, for every $M\models PA$; in fact, it never is.
Now suppose Goldbach is false. Then there is some even $n>2$ such that $n$ cannot be written as the sum of two primes. But this is a bounded statement in terms of $n$: clearly those primes must be $<n$, so the quantifier over primes is bounded; and saying that an element is prime involves quantifying over only numbers less than it. Finally, since $n$ itself is definable by a bounded formula, there is a single sentence $\varphi$, using only bounded quantifiers, which expresses "$n$ is a counterexample to Goldbach". This sentence is true in every model of $PA$, since every model of $PA$ is an end extension of $\mathbb{N}$; so by the Completeness Theorem, $PA\vdash\varphi$. But then $PA$ disproves Goldbach.

Note that this hinges on the fact that Goldbach being false would be witnessed by a single bounded statement about a single natural number (some counterexample). Goldbach being true, however, would not be. So if Goldbach is true, $PA$ might still not be able to prove it.
A statement like the Twin Prime Conjecture is even worse. What is a counterexample to TPC? Well, it's an $n$ such that there is no twin prime above $n$. So the failure of TPC would be witnessed by a single natural number. But the statement that $n$ is a counterexample to TPC involves an unbounded quantifier - " . . . above $n$"! So even if TPC is false, $PA$ might not know. And TPC being true isn't witnessed by a single example, but rather infinitely many - so even though being a twin prime is a bounded statement, TPC being true need not imply that $PA$ proves TPC.
This kind of distinction between mathematical principles leads to a natural hierarchy of sentences of arithmetic - the arithmetic hierarchy - which is intensively studied (along with its extensions and generalizations) throughout mathematical logic:


*

*In computability theory, unbounded quantifiers roughly correspond to the Turing jump, a generalization of the halting problem. This operation is fundamental in computability theory; there is a natural extension of the arithmetic hierarchy to transfinite iterates of the Turing jump, called the hyperarithmetic hierarchy, which in turn is connected with quantifiers in infinitary logic.

*In model theory, the structure of quantifiers in a sentence can determine the kinds of algebraic operations on its models which preserve its truth; see e.g. this. This is essentially the arithmetic hierarchy generalized to arbitrary structures.

*In set theory, the Levy hierarchy is the generalization of the arithmetic hierarchy; quantifiers of the form "$\forall x\in a$" take the place of quantifiers of the form "$\forall n<k$". Additionally, a second-order analogue of the arithmetical hierarchy plays a fundamental role in descriptive set theory, and in the analysis of forcing extensions.

*In proof theory, theories of arithmetic are fruitfully studied by stratifying them into hierarchies of weaker theories via the arithmetic hierarchy.
