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 → bc, b → a, c → 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 criteria), 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?
If I understand your comment correctly, "deciding whether a particular algorithm ..." means an arbitrary algorithm. For a specific algorithm, of course sometimes it is possible to prove that it halts for every input. It's only undecidable if the algorithm is regarded as an input.
