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This part: > Here's how: consider the set of numbers that cannot be described using any finite collection of symbols. Such numbers must exist *because there a
by huitzitziltzin 4y ago
This part:
> Here's how: consider the set of numbers that cannot be described using any finite collection of symbols. Such numbers must exist *because there are only a countably infinite number of numbers that can be described using a finite collection of symbols*, but there are an uncountably infinite number of real numbers.
That’s clearly wrong. The real numbers between 0 and 1 (an uncountable collection) are describable with a finite collection of symbols - the digits between 0 and 9. Each real number is an equivalence class of sequences with the same limit. (It’s been a while since I took real analysis but I believe this is the construction due to Weierstrass?)
- danbruc 4y agoThe author means finite sequence over finite alphabet, i.e. the finite collection of symbols does not refer to the alphabet.
- deleted 4y ago[deleted]
- johnnny 4y agoThe set of numbers that can be written with a finite number of 0-9 digits is countable. That set is called the decimal numbers, and is a subset of the rational numbers which itself is countable too.
- huitzitziltzin 4y agoThe set of numbers which can be written with finitely many members of a finite set of symbols is countable, sure. That doesn’t sound like the claim the author is making in the sentence I quoted but maybe I am being uncharitable?
- johnnny 4y agoIf you allow infinitely long descriptions, then I don't know what's the score. In the case of "descriptions" that amount to (potentially infinite) decimal expansions, I think an argument akin to Cantor's diagonal will show that you can't possibly describe all real numbers (or all numbers between 0 and 1), but I'm not confident I can assert this to be true.