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I came across this amusing question a while ago: "How many computable numbers could a computer compute if a computer could computer all computable numbers?" T

I came across this amusing question a while ago: "How many computable numbers could a computer compute if a computer could computer all computable numbers?" The correct answer (from HN comment section) appears to be 'countably many'. Interestingly computable numbers don't go to infinity: https://en.wikipedia.org/wiki/Computable_number#Not_computably_enumerable https://en.wikipedia.org/wiki/Computable_number#Not_computab... > "While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable." But they do get very very big, i.e. you can theoretically compute numbers so large that: ...even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. https://en.wikipedia.org/wiki/Graham%27s_number https://en.wikipedia.org/wiki/Graham%27s_number

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andrewla 4y ago
> Interestingly computable numbers don't go to infinity: Computable numbers are an unbounded set, so in that sense they do go to infinity, just like the natural numbers go to infinity. They just aren't uncountable. Others and myself have noted in other comments that the Cantor notion of cardinality is not all that interesting or useful compared to the concepts of intuitionalism and constructivism. In particular, as noted in Wikipedia, "almost all real numbers are not computable", yet it is impossible (by definition) to actually produce or approximate one of these numbers through any sort of computational process. This sort of obvious nonsense is why mathematics has very very slowly been edging back from "Cantor's Paradise".

deleted 4y ago

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andrewla 4y ago

> Interestingly computable numbers don't go to infinity: Computable numbers are an unbounded set, so in that sense they do go to infinity, just like the natural numbers go to infinity. They just aren't uncountable. Others and myself have noted in other comments that the Cantor notion of cardinality is not all that interesting or useful compared to the concepts of intuitionalism and constructivism. In particular, as noted in Wikipedia, "almost all real numbers are not computable", yet it is impossible (by definition) to actually produce or approximate one of these numbers through any sort of computational process. This sort of obvious nonsense is why mathematics has very very slowly been edging back from "Cantor's Paradise".