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Wow! This may help with the intuition that typechecking for some type systems is an undecidable problem.
https://beza1e1.tuxen.de/articles/accidentally_turing_comple... lists a few type-related things.

Type inference has been proven to be undecidable in some cases (e.g. with dependent types).

Sorry, by "the intuition that..." I didn't mean "reason to believe (a result that isn't established)" but more like "an aid to understanding (a result that was already proven)".

Thanks for the link!