Axioms - proof of halt
Let $CON(F)$ denote the sentence: "F is consistent".

You can prove $CON(F)\rightarrow \text{ MF does not halt}$, since if MF halts you have $F\vdash G(F)$ and thus by soundness $F\not\vdash G(F)$.

You cannot however, prove this unconditionally (unless your system is inconsistent), since this would mean $F\not\vdash G(F)$ which implies consistency (why?).