Whether or not it does is a matter of which philosophy you choose.

A Platonist may believe that those numbers exist in an abstract space we cannot reach. A Formalist simply defines "existence" in such a way that this is true without worrying about whether they really exist. And a Constructivist denies the existence of things that cannot actually be written down, at least in theory.

This may be clearer to the parent commenter if expressed as "there are only countably many strings drawn from the Unicode alphabet, so if we fix a coding scheme such that a string expresses at most one number in that scheme, then there are only countably many numbers we could have defined. If you try and get around this by saying that the coding scheme is arbitrary so the symbol L can be made to stand for any real, then you have dodged the question of how you specify that coding scheme; there are only countably many coding schemes which can be expressed by strings of Unicode once a coding scheme is fixed, and in particular there are only countably many coding schemes expressible in English".

"Real numbers that cannot be defined with language" is not a well-defined set though. Just like "integers that cannot be described in less than 100 words" is not well-defined (I could reach a contradiction by pointing to the "smallest integer that cannot be described in less than 100 words").

True, but the argument that one set is larger still works.

Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? I'm sure some sort of paradox similar to the one with integers can be constructed here...

In my view, every real number is well-defined and there's nothing controversial about the set of real numbers. If the infinite aspect of it causes some researchers to call it a "mathematical fantasy", so be it, so is literally every other mathematical model we use in our lives.

Your example is faulty.

> I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language?

The "indescribable numbers" are dense in [0,1], and so (if the set exists) the inf of the set of indescribable numbers which are between 0 and 1 is 0. Perfectly describable.

My example is incomplete, not faulty. I left it as a question (does the inf belong to the set?). If the answer is yes, we reached a contradiction. If the answer is no, we have to continue further zooming in to this interval (or some other construction along those lines).

See, I claim that this set is ill-defined, so I can't know its properties like whether or not it's dense, open, closed, Borel-measurable, etc. etc.

You have to tell me what its properties are, and I will come up with a concrete proof that the set in question is ill-defined.

EDIT: After I RTFA'd, this is actually the paradox in section 2.3 of the linked article

> In my view, every real number is well-defined...

How so? The set of definable numbers in any formal langauage might not be clear concept. But you are making a stronger statement.

For any given language, like for instance ZFC, we can say that definable numbers are a countable subset. Hence measure zero.

Then we mean different things by define. I am saying the set R (with all its elements) is an uncontroversial, well-defined construction within ZFC.

I am leaving out any linguistic or Turing-computability aspects out of this, and people try to bring it back in, mixing computability with definability.

For instance, Chaitin's constant is a perfectly well-defined number, albeit uncomputable by construction: https://en.wikipedia.org/wiki/Chaitin%27s_constant

Defining the set of real numbers is very different from defining all real numbers. Yes, Chaitin's constant is defined(with a computable system as a parameter).

But that's the point - we cant produce such a definition for almost all reals.

> Defining the set of real numbers is very different from defining all real numbers.

I'm saying that ^ sentence makes no sense to me, I don't know how to parse it formally. If you start talking about the set of "definable" numbers (not computable, but specifically "definable"), I believe you're gonna run into paradoxes as it's an ill-defined concept, similar (in spirit) to "all integers described under 100 words". In fact, the linked article actually talks about it in 2.3.

> For any given language, like for instance ZFC, we can say that definable numbers are a countable subset. Hence measure zero.

If I can describe a set of objects, then we're all set as far as I'm concerned (mathematically speaking). Being able to efficiently construct individual elements of this set using Turing machines or other computational devices is an orthogonal problem.

Also, I don't think having only countable number of utterances in ZFC precludes you from having well-defined uncountable sets described in that system (quite obviously, for any set S take 2^S which is very well-defined).

The point is straightforward - the fact that you have defined a country on a map, doesnt mean you have defined all its cities and towns. Especially if the number of markers you have are less than the number of towns.

Also, we can talk about definable numbers as long as we choose some specific system which we assume is consistent. So we are talking about numbers which are definable by predicates using the language of ZFC or Peano Axioms.

There is no need to invoke computability, just definability is sufficient. There are lots of definable numbers which are not computable(like Chaitins constant or the real number whose digits encode information about halting of Turing Machines).

But even with this more relaxed constraint, we still dont have enough definable numbers.

If you can describe a set of objects thats fine. But by itself that would be nearly useless. The problem is that ZFC and the like add an axiom that you can identify an element of any described set, and use that to prove further theorems. That makes no sense; its an elimination rule with no corresponding introduction, materializing members of a set from nothing.

There are more people on earth than, say, kings. That's true, even though I can't enumerate all non-kings, and even if the set of non-kings is somehow ill-defined.

Still not sure I follow. Set of non-kings is not ill-defined, not in the same way as set of numbers non-describable by any formal system is.

> Set of non-kings is not ill-defined, not in the same way as set of numbers non-describable by any formal system is.

Um... roughly the same. Is Robert Mugabe a king? Since we didn't give a clear and precise definition of "king" you can't really say.

No, my "ill-defined" means "will lead to contradictions if you look too closely", not "I haven't exactly specified what it means".

A sentence which fails to specify a real uniquely… fails to specify a real uniquely. Borel's talking about numbers which can be defined uniquely: that is, picked out, identified. Your Berry-paradox description doesn't identify an integer.

But if we assume that we can say a sentence either specifies a real number uniquely or does not, the Berry paradox number is uniquely specified.