Simple example of halting-unprovable Turing machine
Take the "current mathematical system" $S$ and an undecidable sentence $P$ for $S$. Consider the Turing machine $T$ which enumerates all theorems of $S$ and halts if it ever enumerates either $P$ or $\lnot P$. System $S$ cannot decide whether $T$ halts.

Note that such a $T$ does not halt.