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CronoDAS comments on No coinductive datatype of integers - Less Wrong Discussion
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I think I recall reading somewhere that you only need first-order logic to define Turing machines and computer programs in general, which seems to suggest that "not expressible in first order logic" means "uncomputable"... I could just be really confused about this though...
Anyway, for some reason or other I had the impression that "not expressible in first order logic" is a property that might have something in common with "hard to explain to a machine".