[chmod775]

> 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]

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]

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]

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]

> 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]

First of all, Pi is not an example of such a number, since it can be defined constructively without requiring physical measurements.

Second, even your proposed mechanism of picking out points would be capable of identification (albeit not computation) in a Turing-complete language. Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points - this could be a metal bar as with the old meter bar, or some other kind of structure / apparatus which has two points in space (at a stable distance) locked in. Crucially though, atomic / quantum configuration of this stabilized apparatus would be encodable in theory, once again providing us with a way to count numbers.

There are of course numbers which are specific to our universe, such as the fine structure constant, that don't have any objective definition without reference to the world. But there are finitely many of these, and they're nameable, just not computable.

[chmod775]

> First of all, Pi is not an example of such a number

It is in its canonical form (i.e. as a numbers, not something that represents it), which is what I said.

> Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points.

No I don't. I could pick two atoms floating through space somewhere and say "This distance, right now."

I could also say: "The distance traveled by this particle in a second."

Either number is likely to be irrational. Either number could then only be approximated in binary or whichever system your prefer - unless it just so happens to be expressible using some named irrational constants.

> There are of course numbers which are specific to our universe, such as the fine structure constant, that don't have any objective definition without reference to the world. But there are finitely many of these, and they're nameable.

I guess we'll have to agree to disagree that there are finitely many of these. I don't believe there's even countably many. And if they're not countable, they're not nameable in any common language, since in those names are countable.

It would be quite surprising to me if the universe was nice enough to limit itself to things that are neatly expressible using using the somewhat limited systems humans use, even though it has already shown that it does contain things that will never be expressible using a finite set of symbols, such as numbers or an alphabet.