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Notes -
This is a classic trick. You see, either one of two cases must be the case:
In either case, L is decidable by a rather trivial Turing machine. That we do not know which one is the case is not an obstacle, a genius might prove either one tomorrow and then any kindergardener could built the TM.
An actually undecidable problem would be L = {x is a Turing machine | x terminates}. Here, we can't neatly split the world into two cases and just profess ignorance about which world we live in.
(Original thought: it does not feel like the outcome could be independent of whatever axiomatic system you use -- pi's digits being very computable no matter if you like ZF+C or plain ZF or Peano or whatever. (Provable theorems about the digits might be more elusive, though.) However, I would not claim that this is obvious. After all, BusyBeaver(8000) eludes ZF, when my naive assumption would have been that surely, the termination of a TM should be the same under any axiomatic system which can define it.)
D'oh, I should've realised that. Thanks for the handholding.
(I'm with you regarding the intuition.)
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