Say we have some theory $T$ such that $Th(A_E) \subseteq T$ where $A_E$ are the axioms of arithmetic. How do I show that
(1) there are sentences $\varphi_1$ and $\varphi_2$ such that $Th(A_E) \vdash \varphi_1 \leftrightarrow \psi(\varphi_2)$ and $Th(A_E) \vdash \varphi_2 \leftrightarrow \neg \psi(\varphi_1)$, where $\psi(\alpha)$ represents that $\alpha$ is provable from $T$.
(2) That $T \vdash \varphi_1 \rightarrow \neg \rho$ where $\rho$ expresses that $T$ is not consistent.
Here are my thoughts: (1) If we have that we can prove that $\varphi_1$ is unprovable and call that statement $\varphi_2$, then we see that $\varphi_1$ represents the statement that it is unprovable that we can prove $\varphi_2$ which seems to follow. I'm not sure how to formalize this reasoning and every attempt I make seems to result in my relying on the existence of either $\varphi_1$ or $\varphi_2$.
(2) If $\varphi_1$ and $\varphi_2$ are independent of $T$ (that is that we can neither prove nor refute them, which seems to be the case), then I don't see why the consistency of $T$ is compromised.
Any thoughts would be much appreciated.