As a non-mathematician, I’m confused why so many people think the conjecture (whether true or false) is provable within PA. To me, it seems like something that would be very nicely just right outside the boundary of PA’s capability, sort of like how proving all Goodstein sequences terminate requires transfinite induction up to ε_0. Add that to the fact that the Collatz Conjecture seems to fall in the same “category” of problem as the Turing machines that the Busy Beaver project is having a hard time proving non-halting behavior of, and the heuristic arguments all seem to point to: Collatz is independent of PA.
But I’m interested in hearing the counterarguments that Collatz likely is provable within PA and why this would be the case.
The Collatz conjecture is "obviously" (i.e. probabalistically) true, so it's frustrating to not be able to turn that into a proof. PA doesn't matter, there's no known approach to proving it with stronger theories either. Of course many other propositions like the twin prime conjecture are in the same situation.
Goodstein's theorem by contrast is obviously provable in slightly stronger theories than PA, and it involves a fast-growing sequence which suggests it's out of weaker theories' reach. In fact it encodes ordinals up to eps_0 in a natural way, so its equivalence to CON(PA) is unsurprising. The Collatz conjecture is nothing like that. It's beguilingly simple by comparison.
But I’m interested in hearing the counterarguments that Collatz likely is provable within PA and why this would be the case.