[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.