Q:

Language that describes all real numbers

According to Wikipedia:

Suppose that in a mathematical language $L$, it is possible to enumerate all of the defined numbers in $L$. Let this enumeration be defined by the function $G\colon W\to \mathbb R$, where $G(n)$ is the real number described by the $n$th description in the sequence. Using the diagonal argument, it is possible to define a real number $x$, which is not equal to $G(n)$ for any $n$. This means that there is a language $L'$ that defines $x$, which is undefinable in $L$.

Does it mean there exists a language that can define all real numbers, produced by the diagonal argument?

A:

Well, it depends what you mean by "language." Any countable language will fail to describe every real number, by that argument. However, if we allow uncountable languages, then of course every real can be described by such a language - e.g., a language containing a symbol for each real number.

An important aspect of this answer is that I am talking about the "real" universe (whatever that means). There are models of set theory in which every element - not just every real! - is first-order definable in the language of set theory; such models are called "pointwise definable" and have been studied by Joel David Hamkins and others. Note that such models are necessarily countable.

A:

This is a delicate issue. What do you mean by "describe"?
Can you describe $\pi$ using just the field operations on $\Bbb R$? No. You cannot. But you can, if you allow using $\leq$ and $\sin$ (it is the least positive root of $\sin$).
You can describe $\pi$ if you extend your universe to allow sets of reals, and the only set of real you've added is $\{\pi\}$.
So what exactly does it mean "describe all the real numbers"? This can take two meanings.

We have a language $\cal L$ and the real numbers are a structure to this language somehow. Can every element of $\Bbb R$ be definable using an $\cal L$-formula?
The answer to that is yes, assuming we allow a sufficiently large language. If $\cal L$ is simply continuum many constant symbols, and the structure of $\Bbb R$ interprets each real as a unique constant symbol, then you can easily describe each real number. If the language is countable, then the answer is of course negative.
The answer is also positive if you're using a logic which allows infinitely many conjunctions, and in a language where you can define all the rational numbers. Then you can just describe each real number by saying which rational numbers are smaller and which are bigger.
We can allow a much much larger universe in which we want to describe the real numbers. So for example you can ask, in the set theoretic universe, is it possible that every real number can be defined by a unique formula? The answer to that is in fact positive. It is possible to have a universe of set theory where every set is definable. The catch is that the definitions are not objects of that universe, but rather they live in a lovely place called the meta-universe. But this requires better understanding in both set theory and logic, so I'll just leave this as an anecdote.

