Then you should be able to come up with a function that assigns a natural number uniquely to each real number.
Of course if you tried that I could immediately name you a real number, or a pair of them, for which your rule doesn't work.
I can give you a function that will never output the same natural number on any two real numbers unless those two real numbers are equal.
My function outputs the binary value of the Unicode encoding of the description of the number that you provide.
Go ahead, name me a pair of real numbers for which my rule doesn't work.
That is false - you're actually unable to name these numbers for which the rule doesn't work. Instead you're just insisting we accept on faith the existence of these unnamable numbers.
See https://news.ycombinator.com/item?id=27859355 for why.
It is obvious that all expressions in the English language can be ordered from smallest to largest and lexicographically, which makes these expressions trivially countable. We can thus assign natural numbers to real numbers by assigning numbers to their expressions in a natural or formal language, which will of course include infinitely many expressions that are just nonsense descriptions and infinitely many expressions that map to the same real number. These expressions will also include any possible expressions of Cantor's or other diagonalized numbers. In such a sense then, we can trivially "count" the real numbers unless we hold the philosophical view that there are real numbers that are not expressible. This is where it becomes a question of philosophy of mathematics, not mathematics proper.
You can of course object that what you meant by "assigns uniquely" is an unambiguous 1:1 mapping and that including any number of nonsense descriptions misses the point. In that case giving a "rule doesn't work" because the diagonalized number always escapes the proposed system of counting the numbers, but only because the diagonalized number is allowed to 'parasitically' depend on the totality of the system, but is excluded from the system (or else it would diagonalize itself and become ambiguous at that particular decimal place). This particular viewpoint is tied to a particular philosophical position, however, and not all positions in the philosophy of mathematics will agree with it.
This all might seem trivial or even nonsensical (as philosophy of mathematics so often appears), but I merely want to point out that the 'uncountability' of the real numbers is not a consequence of the set of the natural numbers being 'too small' to hold all the real numbers, because they are 'large enough' to assign numbers to all possible descriptions all real numbers that will ever be expressed in language. Uncountability is a consequence of a view that restricts Cantor's diagonalized number from the set of the countable number but still considers this diagonalized number to be a real number (which again is only unambiguously defined if it is not allowed to diagonalize itself). There are however other possible philosophical viewpoints which either include the diagonalized number in the set of countable numbers (at the cost of including ambiguous or paradoxical numbers) or reject the view that Cantor's diagonalized number should be considered to be a real number in the first place.
tl;dr: Yeah, you can always name a real number for which a particular counting rule does not work, but only as long as there is agreement regarding the philosophical underpinnings. Most mathematicians can probably be considered platonists and from their standpoint the real numbers are obviously uncountable, but that is by no means true for all positions in the philosophy of mathematics.
This doesn't work because not all real numbers have expressions in a natural or formal language. This is easily shown by an obvious variation on Cantor's diagonal proof, applied to your lexicographically ordered list of expressions in any natural or formal language.
> In such a sense then, we can trivially "count" the real numbers unless we hold the philosophical view that there are real numbers that are not expressible. This is where it becomes a question of philosophy of mathematics, not mathematics proper.
I'm not disagreeing with your interpretation of Cantor's diagonal proof, I'm merely pointing out that this interpretation depends on a very specific philosophical view of mathematics, namely the platonist view that the real numbers exist independently from their expressions in any natural or formal language and that it makes sense to say that there are real numbers that are not expressible.
And yeah, nearly all working mathematicians will agree with this view and from their perspective the real numbers are uncountable, period, and you are right that what I sketched "doesn't work".
But I think it's important to remember that there are or could be alternative philosophical views of mathematics that lead to a different interpretation, which will reject not the mathematical validity of Cantor's diagonal proof, but rather its usefulness or relevance. After all, how can you convince someone that there are real numbers that are not expressible? By their very nature they cannot be practically used in any calculation, so how could you convince someone who is not convinced by this philosophical assumption of Cantor's diagonal proof?
In other words, Cantor's diagonal proof cannot prove that there are real numbers that are not expressible, because the proof only makes (philosophical) sense if you accept this viewpoint in the first place.
You could say the same about just real numbers, which can also be ordered from smallest to largest. This definitely not 'trivially' implies countability.
Edit: I edited my post shortly after posting from "Humor me and count out the first two real numbers".
> Humor me and count out the first two real numbers in English in lexicographical order.
Probably "one" and "six" (then "ten" and "two", followed by all 4-letter expressions of real numbers), unless you can think of an English expression with a length of three letters or less that would occur before "one" and "six" lexicographically. If you consider "pi" or "e" to be descriptions of real numbers then clearly these would occur before "one" and "six".
But of course you can also pick an arbitrary universal Turing machine or other suitable formalism, pick an encoding as bit strings and order expressions in such a formal language according to their bit strings. Ordering expressions in the English language is only the less formal counterpart.
(If your point is that ordering expressions of real numbers in English is far from unambiguous without first agreeing on a dictionary of valid words and on rules of what can be considered an expression of a real number in more than one word then of course I would agree. But what I'm driving at is not that it's easy to unambiguously count out the real numbers in English, of course it's not, but rather that any kind of natural or formal language expressions, by being recursively enumerable, are "numerous" enough to be a countable set of expressions. To say that the real numbers are uncountable is to accept the view that there are real numbers that are not and can never be expressible in language, which is a view that only makes sense against the backdrop of a very specific philosophical framework. One that is definitely accepted more or less implicitly by most working mathematicians, but not the only possible one. And this philosophical framework cannot itself be justified or grounded by a mathematical argument such as Cantor's diagonal proof.)
---
Reply to the parent after edit:
> You could say the same about just real numbers, which can also be ordered from smallest to largest. This definitely not 'trivially' implies countability.
No, what I meant was that by ordering expressions in the English language or other suitable formal languages first from smallest string to largest string and within these groups lexicographically, you can enumerate all the expressions in such a language, which makes them trivially countable.
Of course this does not give you a 1:1 mapping from expressions in a language to real numbers, but it is meant to illustrate that the real numbers are not uncountable because they are 'too numerous' to be counted by the natural numbers, in the sense that a box is not large enough to hold a collection of things. 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.
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?
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).
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.
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.
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.
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.
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.
I guess that you've never heard of the Heisenberg Uncertainty Principle?
You can say that. But it isn't actually well-defined.
This isn't a coherent sentence. A real number is a cauchy sequence of rational numbers. There is no one canonical sequence that defines Pi.
> 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."
That process of picking out and localizing the atoms (which are not localized to specific points) would entail performing a measurement with a macroscopic apparatus. At that point, the configuration of the apparatus can be encoded.
If you're careful, it can't be shown at all.
As I already pointed out, if classical mathematics is consistent, then constructivism must be as well. Therefore if you think that you've found a logical flaw in constructivism, the mistake must be in your own thinking.
What does "to exist" mean for number which can not be written down as some formula (in broad sense of this word) in formal language?