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Nornagest comments on Could Robots Take All Our Jobs?: A Philosophical Perspective - Less Wrong
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Humans can't "solve" it either, in that sense. We can pattern-recognize that some programs will halt or not halt, but there exist huge spaces of programs in between where we would be just as helpless to give a yes or no answer as any computer program.
I'm not sure what this should be considered evidence of, but somehow it seems relevant. ;-)
For that matter, it's perfectly possible to build algorithms that can accurately tell you whether or not some systems will halt -- certain types of infinite loop are easily machine-detectable, to give one simple example. It's doing it in the general case that's impossible.