[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).

[enugu]

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.

[kazagistar]

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.