In an online discussion on Turing machines and decidability recently, I blatantly theorized that any problem about a specific single Turing machine must be decidable, the question of undecidability only arises when we are talking about a class of a problem.
As a rebuttal to the original argument someone replied that if we have a Turing machine and we give it an input w it’ll be undecidable to say will it halt or not and was proven by Turing himself.
Is this a correct argument because skimming through wikipedia it looks that’s not what Turing actually proved
Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist
What he proved was that no general solution exists for all possible program-input pairs and not that we can’t have a specific algorithm to solve this on a single machine (e.g if we take a halting Turing machine a simple solution would be to say yes for all inputs).
In support of my original statement look at the PCP, no general solution exists but if we are given two string we can sure brute-force them by looking at all possible subsequences and tell whether a correspondence exists or not.
So what I am asking here does my original statement hold weight and what exactly did Turing proved.