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> Probably "one" and "six" You missed infinitely many numbers between those two. For example "one thousand" and "one dot/comma three". Once you've worked that

> Probably "one" and "six" You missed infinitely many numbers between those two. For example "one thousand" and "one dot/comma three". Once you've worked that all out, can you now tell me which natural number you assigned to "six" in your lexicographical order? Edit: Ah. Took me a moment to realize you are ordering by number of characters first. > They are uncountable because we accept the view that there can be real numbers that are not and cannot be expressible in language, which is a platonist view that is open to philosophical critique. So similar to R \ Q? Or the same?

skoodge 5y ago
Oh, I see what the issue is. I wrote "smallest to largest and lexicographically", meaning first from smallest to largest (as measured by the length of the string) and then within each group lexicographically, which is the usual way of giving an enumeration of such expressions as far as I know. Of course you cannot enumerate these expressions if you expect a solely lexicographical ordering. In any case, English was just an example, you can basically pick any Turing-complete language that is recursively enumerable and count its expressions by considering its bit string encodings as natural numbers. Edit: This is also why the computable numbers are countable, but not computably enumerable (because figuring out which expressions correspond to real numbers is equivalent to the halting problem).
- chmod775 5y ago
  You've convinced me that R without all numbers that are not expressible in typical languages is countable at least. Though I'm not convinced that numbers that are not expressible don't exist. I could for instance say "the length of this line", which may very well have a length that is not expressible in a language that uses a finite set of symbols (hah, that's why your expressions are countable, of course!). Consider a language however that instead of numbers simply uses sounds of the appropriate length, or draws lines of proportional length (assuming of course you could do this precisely). With that language you could express every real number, and you could express numbers that turing machines* or English can't. *Unless programmed in that language.
  - skoodge 5y ago
    That's an interesting thought experiment, though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language. To say that there are inexpressible real numbers is to say that there are numbers that are not rational but which can never be practically used or accessed either, at which point they become kind of like an invisible and unnoticeable unicorn: it is certainly possible to believe in its existence, but such a belief is quite different from the belief in the existence of practically useful real numbers such as pi. I am not trying to convince anyone that inexpressible real number do or do not exist, but I think it's worth noting that these issues quickly cross over into the realm of philosophy, where it's not possible to justify a particular conviction by appealing to firm mathematical or practical reasons. Nothing wrong with that, of course. Personally, I'm content with what is expressible in language and I consider mathematical concepts going beyond this boundary of expressivity as inessential to my own personal use, though I can certainly see that these mathematical calculi can be of interest to mathematicians on their own.
    - chmod775 5y ago
      > though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language Consider the canonical form of Pi. You can't express it accurately in English, but you can have a distance of Pi length in the physical world, so you could express Pi by that distance. Now we can refer to Pi in English because it is tied to a concept which we can describe and because we can assign a name to it. If I picked two arbitrary points in the physical realm, then due to the distribution of real numbers there's a good chance you wouldn't be able to express the distance accurately in a language that uses a finite set of symbols/sounds to express numbers. I'm convinced such 'numbers' exist however, Pi exists after all, and it happens to be just one we assigned a name to because it is of note.
      - ummonk 5y ago

skoodge 5y ago

Oh, I see what the issue is. I wrote "smallest to largest and lexicographically", meaning first from smallest to largest (as measured by the length of the string) and then within each group lexicographically, which is the usual way of giving an enumeration of such expressions as far as I know. Of course you cannot enumerate these expressions if you expect a solely lexicographical ordering. In any case, English was just an example, you can basically pick any Turing-complete language that is recursively enumerable and count its expressions by considering its bit string encodings as natural numbers. Edit: This is also why the computable numbers are countable, but not computably enumerable (because figuring out which expressions correspond to real numbers is equivalent to the halting problem).

chmod775 5y ago
You've convinced me that R without all numbers that are not expressible in typical languages is countable at least. Though I'm not convinced that numbers that are not expressible don't exist. I could for instance say "the length of this line", which may very well have a length that is not expressible in a language that uses a finite set of symbols (hah, that's why your expressions are countable, of course!). Consider a language however that instead of numbers simply uses sounds of the appropriate length, or draws lines of proportional length (assuming of course you could do this precisely). With that language you could express every real number, and you could express numbers that turing machines* or English can't. *Unless programmed in that language.
- skoodge 5y ago
  That's an interesting thought experiment, though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language. To say that there are inexpressible real numbers is to say that there are numbers that are not rational but which can never be practically used or accessed either, at which point they become kind of like an invisible and unnoticeable unicorn: it is certainly possible to believe in its existence, but such a belief is quite different from the belief in the existence of practically useful real numbers such as pi. I am not trying to convince anyone that inexpressible real number do or do not exist, but I think it's worth noting that these issues quickly cross over into the realm of philosophy, where it's not possible to justify a particular conviction by appealing to firm mathematical or practical reasons. Nothing wrong with that, of course. Personally, I'm content with what is expressible in language and I consider mathematical concepts going beyond this boundary of expressivity as inessential to my own personal use, though I can certainly see that these mathematical calculi can be of interest to mathematicians on their own.
  - chmod775 5y ago
    > though I'm not sure how using "sounds of the appropriate length" or "lines of proportional length" would get you more than the rational numbers, which are already countable and thus fully captured by any Turing-complete language Consider the canonical form of Pi. You can't express it accurately in English, but you can have a distance of Pi length in the physical world, so you could express Pi by that distance. Now we can refer to Pi in English because it is tied to a concept which we can describe and because we can assign a name to it. If I picked two arbitrary points in the physical realm, then due to the distribution of real numbers there's a good chance you wouldn't be able to express the distance accurately in a language that uses a finite set of symbols/sounds to express numbers. I'm convinced such 'numbers' exist however, Pi exists after all, and it happens to be just one we assigned a name to because it is of note.
    - ummonk 5y ago