[upquark]

"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").

[FabHK]

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

[upquark]

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.

[Smaug123]

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.

[upquark]

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

[enugu]

> 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.

[upquark]

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

[enugu]

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.

[upquark]

> 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).

[FabHK]

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.

[upquark]

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.

[mcherm]

> 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.

[upquark]

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

[Smaug123]

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.

[zodiac]

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.