So your argument is that there's no GI because GI can't be defined? Hence your statement 'there's no GI' has no truth value. You don't know if it halts.
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TLDR: GI is a panchreston.
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A few decades ago you could say the same thing about biological life.
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No need to go that far into the misadventures of the first taxonomists (species, a core concept of biology is not free of problems), since there are good modern arguments on the problems of categorical definitions of cognition.
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A lot of definitions are human centric. When someone says Poker or Go is solved, that's a technical fiction; actual truth of it is merely negligible probability that a human wins against the machine.
Similarly, AGI would have comparable task breadth*depth ability to human
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And if one wishes to be specific about general, noting that humans aren't that general then it would be vs say, 100 humans.
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One could further argue that a human in general because of their interaction with 100 other humans.
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Right, without being embedded within such interactions, humans wouldn't be so capable but, full capabilities wouldn't be located in any one human, they'd be distributed.
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Furthermore, one can say the same for anything biological. Its full capabilities are always within the context of what it's embedded in.
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We have this illusion perhaps because we are the dominant species on earth.
The day when there is a species (AI) more dominant than us, which will discover problems that we can't even imagine, then the GI will change its definition.