"This sentence has no proof" (and add all the difficult bits about formal systems, what you mean by proof, and the containing of enough complexity to model arithmetic) is a paradox, and it's led to some very interesting mathematics. (I observe that Goedel said that any true paradox could be used as the basis for an incompleteness theorem.) Yes, it's also led to a lot of cranks spouting huge amounts of nonsense, but that just proves that people will insist on misunderstanding things, and then loudly demonstrate their lack of understanding.
Paradoxes in the true mathematical sense are useful in mathematics, and I thought that many here on HN would find that they sufficiently provoke their intellectual curiosity to be regarded on topic.
However, clearly you disagree, so clearly we disagree. I found the submission interesting, the discussion less so, and wish the inappropriate comments had been flagged/killed to allow the interesting and contentful discussion to flow. That's not what happened.
People complain that HN has jumped the shark, and the submissions are of poor quality. Personally I think it's the inappropriate comments that are dragging HN down, the insistance by so many to add largely content-free and insight-free comments. And yes, I admit that I've done it too.
PG says that the greatest threat to HN isn't the poor quality of submission, but the poor quality of comment. Perhaps submissions should die less often, and comment be killed more often.
Perhaps not. I don't have solutions, but I do know that this submission was a paradox, and that some true paradoxes are of interest and use in formal mathematics. Others aren't - it would be interesting to know if this is the former, and that's what I'll be exploring with some of my colleagues over the next few days.