Peano's theory of arithmetic and Gödel's 1st Incompleteness Theorem

Question

Let $\mathcal{N}$ be Peano's 1st order theory of arithmetic and $\mathscr{A}$ it's standard model (which we assume exists). Infer from Gödel's 1st Incompleteness Theorem that there exists a closed well founded formula say $B$ of $\mathcal{N}$ and a model $\mathscr{W}$ of $\mathcal{N}$, such that $B$ is ture in $\mathscr{A}$ and $B$ is false in $\mathscr{W}$. Any help is really needed and app recited. Thanks in advance.

Answer 1

Take a canonical Gödel G sentence for PA i.e. a wff that "says" of itself 'I am unprovable in PA'. Ask yourself:

1. Is G provable from the axioms of PA?
2. Is G true in every model of the axioms of PA?
3. Is G true at least in the standard model of PA?
4. What can you deduce from your last two answers?

Hint: for one of these answers you appeal to Gödel's completeness theorem for first-order logic, and the fact that PA is a first-order theory.