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The proof is simple: there are countably many names, but uncountably many real numbers. Hence, some real numbers must be unnameable.
by nofriend 1mo ago
The proof is simple: there are countably many names, but uncountably many real numbers. Hence, some real numbers must be unnameable.
- goodmythical 1mo agoAre there countably many names? Countably sayable, perhaps, but I don't recall seeing any limitation to word length in the spec. See: "Below is the full 189,819-lettered word for 'titin':" at https://cw39.com/wp-content/uploads/sites/10/2020/09/longest-word.pdf https://cw39.com/wp-content/uploads/sites/10/2020/09/longest... as an example. Surely if we can have a 189,819 lettered word we can have a million billion trillion lettered word or an uncountably lettered word. They'd be used in the same way as the really long numbers in that we'd give them some other handier name that collides when not given context. e.g. In spoken language pi/pie are often confused if the conversation does not already have a mathematical context and no one says 3.1415926535... conversationally just as no one uses the lenghtier version of chitin and no one would use the uncountably long name for some uncountably long number.
- nofriend 1mo agonames have to be finite in length. i think that's pretty obvious
- goodmythical 1mo agoI don't see how that's any more obvious than the suspicious claim that numbers can have only so many digits.
- nofriend 1mo agoA number is not in the first place its digit sequence. A number like pi is in the first place the ratio of a circle's diameter and its circumference, and only incidentally a certain (infinite) decimal expansion. A name is in the first place something you say, hence the thing you say has to be (at least theoretically) sayable.