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wmorgan comments on No coinductive datatype of integers - Less Wrong Discussion
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Agreed. There's an even easier way to confound the machine, really: when it asks you for the bit string, walk away from the computer!
Note that there is no way to feed infinite input to a Turing machine. The halting problem isn't demonstrated by a program of infinite length. It's demonstrated by a finite program that resists static analysis. The analogous integer is not the one represented by an infinite bit stream, but it might be something like the Berry number.