TITLE: Interaction of completeness and second incompleteness theorems
QUESTION [5 upvotes]: So I was reading the Wikipedia article on Godel's completeness theorem, the section on its relation to completeness. It says that completeness gives the existence of a model of arithmetic $\mathcal M \models PA$ where $\mathcal M \vdash \neg \text{Cons}(PA)$, but that the Godel number of the any contradiction is non-standard. Unfortunately, Wikipedia gives no citations for this claim.
Why does completeness imply the existence of this model? How does the Godel number of the contradiction being non-standard avoid actually proving that PA is inconsistent? What does "consistent when viewed from outside" mean? How does that work out?

REPLY [6 votes]: Simplifying a little bit, Gödel's Completeness Theorem for first-order logic (with one of the "usual" proof systems, i.e. logical axioms+rules of inferences) says that : 


$\Gamma \vDash \varphi$ iff $\Gamma \vdash \varphi$,


i.e. all the logical consequences of a set of sentences are provable with the (logical) rules and axioms.
If we apply it to $\mathsf {PA}$, we have that $\Gamma$ is the set of f-o Peano's axioms.
The above theorem says that all f-o formulae of $\mathsf {PA}$ that are provable from Peano's axioms are logical consequences of the axioms, i.e. they must be true in all models of the axioms.
Godel's Incompletenss Theorem says that $\mathsf {PA} \nvdash Cons(\mathsf {PA})$.
But, if we assume $\mathsf {PA}$ consistent, and this assumption is necessary for the proof of G's Incompletenss Th, we have that the formula $Cons(\mathsf {PA})$ is true.
$Cons(\mathsf {PA})$ being unprovable from f-o Peano's axioms, we have that it is not a logical consequence of the axioms.
And if it is not a logical consequence of the axioms, it is not true in all models of the axioms, i.e.it must be false in some of their models. 
But we know that it is true in the "standard" model $\mathbb N$; thus, it must be false in some "non-standard" model.
Then, Arthur's argument follows.