I have just read about Terence Tao's approaches to Collatz conjecture, and I have a stupid question.
If binary representations of numbers in the hailstone sequence for Collatz conjecture can be written as a $2$-tag system ($a \to bc$, $b \to a$, $c \to aaa$), and $m$-tag systems are Turing complete for $m>1$, and Collatz conjecture is a sort of halting problem in this framework (there is a halting criterion), and "deciding whether a particular algorithm for Turing machine will/won't halt on every input" is a totality problem, which is also undecidable — can a proof of undecidability of Collatz conjecture be approached this way? What is the major problem with this line of thought?