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OP is about natural numbers, but if the challenge is generalized to computable ordinal numbers, then it actually becomes a very interesting measurement in the a
by xamuel 7y ago
OP is about natural numbers, but if the challenge is generalized to computable ordinal numbers, then it actually becomes a very interesting measurement in the area of AGI. There's no bound on how big of a natural number any particular AGI could name (because if an AGI can name N, then it can name N+1, and by repeating that process, all the naturals are covered). But the set of codes of computable ordinals is a non-computable set, so no AGI could ever name them all, every AGI necessarily would have an upper limit on how large of a computable ordinal it could ever name.
See my slides on the subject here: https://semitrivial.github.io/MeasuringIntelligence2019.pdf https://semitrivial.github.io/MeasuringIntelligence2019.pdf
- lonelappde 7y agoHow is this relevant to AGI?
- xamuel 7y agoHow would you measure the intelligence of an AGI? Say you've got HAL in one corner of the room and The Terminator in the other, and you can ask them questions and ask them to perform computations, and they'll obey you. How can you make sense of which one is smarter? A naive attempt would be to ask them to name the biggest (natural) number, but that's no good: HAL says 1000, Terminator says 1001, but of course HAL also knows 1001 is a number so Terminator only wins by dumb luck. We can salvage the idea though by switching natural number for computable ordinal number, and instead of having them name a single number, have them enumerate computable ordinal numbers indefinitely. The set of codes of computable ordinals is Turing non-computable, so neither machine will succeed in enumerating all of them, and by comparing the size of the ordinals enumerated by the two machines, you get an elegant, parsimonious, not-too-contrived notion of which machine is (mathematically) smarter. Read the slides I linked!