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hey math nerds is (something similar to) this true:
if a dense subset of the reals is decidable by any finite algorithm (for ~any reasonable formulation of "finite algorithm"), then the neighborhood of any point has measure 1 or 0
it seems true? but i don't know what vocabulary to search for to learn more
@fae I dunno, but typical (finite) algorithms have problem with general real numbers - too many of them aren't computable. Probably one needs work only with the computable numbers.
Next step is to use Rice's theorem (https://en.wikipedia.org/wiki/Rice%27s_theorem) to argue that the only decidable sets of computable numbers is either set of all of them, or empty set.
@fae Do I understand the 2nd part correctly? ... I want to interpret it like "the intersection of any neighborhood of any point and this set has measure 1 or 0."