There is no such thing as "the shortest program for which the halting property is uncomputable". That property is computable trivially for every possible program.
The halting property is semi-decidable: if a program halts, then you can always trivially prove that it halts, you just need run it. If a program does not halt, then sometimes you can prove that it doesn't halt, and sometimes you can't prove anything.
For any programming language, there exist a length such that you can compute the halting property for all programs shorter than that. At length L_max, there will be program Bad_0, which:
You can never prove that any string S has Kolmogorov complexity K if K > L_max, as it would imply that you proved that Bad_0 doesn't halt, or at least doesn't emit any symbol which is not a prefix of S.
Since there are only finitely many strings with complexity up to L_max, we can only compute Kolmogorov complexity, for finitely many strings.
It is also easy to make up artificial languages in which Kolmogorov complexity is computable for an infinite subset of all possible strings.
If the language is Turing-complete I don't think this is possible. If you think that an arbitrary string S has complexity K, how can you prove that there exist no program shorter than K that computes S?
[retracted]
I know this reeks of witch-hunting, but... I have a hunch that u/Eugine_Nier is back under the guise of u/Azathoth123. Reasons:
I don't have an axe to grind against the guy, I've only spoken to him a couple of times and didn't notice any particularly large karma hits afterwards, I just really dislike it when someone skirts the rules like that. Disruptive users evading permanent bans never helped any community ever.
Obviously I'm posting this here because I think a moderator should look into the matter. Usually I would be posting a disclaimer of some sort, apologizing in advance to Azathoth123 for attacking his standing with slanderous accusations if this turned out not to be the case. Well, I won't. The more I look into the matter, the more confident I get that they're the same person. Azathoth, if you're reading this and you're not Eugine_Nier, then I strongly advise you go search for your twin brother, I think you'll get along very well. Seriously, I'm saying this in good faith. You have a suspiciously great deal of things in common.
If retributive downvoting is (still) a concern (if not, then disregard this paragraph): I'd like to request, if such a thing is possible, that a mod karma-blocks me until the issue is over, so as to not incur undeserved downvotes (it would also mean I'd get no upvotes). In turn, I promise not to abuse the system by spamming the boards with garbage without consequences, but then again given my history so far on LW I don't think that such an abuse should be expected from me. For the record, I could have made a throwaway account just to say this, and not risk being karmassassinated, but 1) a zero karma account has no credibility and 2) for signalling reasons I prefer to put my money where my mouth is.
P.S. I only made this announcement its own post because the latest open thread was about to "expire".