Simple example of halting-unprovable Turing machine Is there a simple example of a Turing machine $M$, such that whether $M$ halts or not on the empty input cannot be proved within the current mathematical system?

Specifically, I'm curious whether there exists an example simple enough that the state transition of the Turing machine can be drawn as a reasonably-sized diagram.
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.