What is vulgar materialism and what does it have to do with this? (I realise this is a shitpost but maybe there's something to learn anyway)

This is essentially @gro-tsen.bsky.social's answer reformulated in a simpler way: consider the set of programs which syntactically have one of the two forms

fun n => if n = 0 then k else …
fun n => if n = 0 then loop () else …

OK, so let T be the set of data ⟨n,e⟩ where e is a Turing machine and n∈ℕ such that φ_e(0) finishes in ≤n steps. Consider the program h that takes ⟨n,e⟩∈ℕ and x∈ℕ and runs φ_e(x) except that if x=0 its execution is allowed to run for at most n steps, else h enters an infinite loop. … •1/6

Right, this was asked in the comments of my answer and I answered it negatively.

cs.stackexchange.com/a/172967

OK, I'll edit my answer.

Amazing, thanks! Do you want to post this as an answer or should I edit mine?

Also note that this might depend on the choice of admissible numbering φ. The question was more specifically about Turing machines.

(That is, there is a partial computable function ℕ → bool that restricts to the halting function on S.)

A positive answer to this would answer my question negatively, but the actual question is: is there a decidable subset S ⊆ ℕ that meets every equivalence class of ~, and for which the halting problem on the empty input is decidable.

(Well, one more question; the second one was just me being silly.)

I did a long twitter thread on Chao's Jabberwocky translation and his invented Chinese characters, including an image of his annotated proof of the lost 1932 edition of 《走到鏡子裏》("Through the Looking-Glass") prepared by the Commercial Press in Shanghai but blown up by the Japanese.

I've been obsessing over this ever since I moved to Sweden. It seems like the phenomenon is called [Viby i](sv.wikipedia.org/wiki/Viby-i) (named after a place, not the English word!) but none of the Swedish people I've talked to said they'd noticed it. It seems very widely spread these days.