The Consistency of Arithmetic I believe that within ZFC (or maybe even a weaker subset of ZFC) there is a proof of $\mathbb{N}=(\omega,+,.,<,0,1)\models{PA}$. What would be a standard reference for this?
 A: The main difficulty here is that there is a difference between showing

For each axiom $\Phi$ of PA, ZFC proves $\mathbb{N} \vDash \Phi$.

and showing

ZFC proves $\mathbb{N} \vDash \text{PA}$.

The difference is that in the former case the quantifier over formulas is in the metatheory, while in the latter case the quantifier over formulas is in the object theory (which is ZFC in this case). 
To prove the second statement, the key point is that ZFC proves the following

For each coded formula $\phi(n)$ in the language of PA, with one free variable, the set $\{m\in \omega: \mathbb{N} \vDash \phi(m) \}$ exists. 

This in turn relies on the fact that ZFC is able to construct the satisfaction relation for $\mathbb{N}$. Once you have these things, and you know that ZFC proves $\omega$ is well founded, it is straightforward to verify in ZFC that each instance of the (infinite) induction scheme of PA holds in $\mathbb{N}$.  It is also straightforward to verify in ZFC that $\mathbb{N}$ satisfies each of the finite number of remaining axioms (which make up $\text{PA}^{-}$)
So the main source of difficulty is just with formalizing into ZFC the fact that the single second-order induction axiom
$$
(\forall X \subseteq \mathbb{N})( 0 \in X \land (\forall n \in \mathbb{N})[n \in X \to n+1 \in X] \to (\forall n \in\mathbb{N})[n \in X]).
$$
implies every instance of the first-order induction scheme. 
