Are there real numbers that have no finite description?

Question

Here's my argument:

Descriptions of finite length are countably infinite (we can enumerate through all descriptions of max length 1 with 1 distinct symbols, then max length 2 with 2 distinct symbols, etc.)

Some finite length descriptions specify a single real number including the axioms/definitions needed to define the real number.

Therefore the real numbers that have a finite length description are a subset of all finite length descriptions.

Since there are uncountable reals, then some reals have no finite length description.

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Someone may argue, we cannot define which of those descriptions correspond to real numbers and which do not.

Agreed - due to Godel's incompleteness theorem, some of those descriptions will correspond to real numbers that cannot be proven to be real numbers from the axioms and definitions in the description.

Therefore, given a finite description of axioms and definitions and a candidate real number, we cannot decide whether or not it's a real number from the axioms and definitions in the description.

The enumeration of defineable real numbers, then is not defineable.

However, every defineable real number is an element of descriptions of finite length.

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A similar question is here: Are there real numbers that cannot be uniquely expressed with a finite number of symbols?

I don't understand this part of this answer:

Externally, it is possible (but not necessary) that every real number can be uniquely expressed in the same metalanguage you use to describe the set theory you're using. Here is a related question asking whether, not just each individual real number, but every set can be defined.

Apparently there's a model in ZFC that can specify each set - and thus real number. How does that work with my argument? Does that model require an infinite length description? If so, then to fully describe those real numbers, you need to fully describe the model of ZFC, which requires an infinite description. That's compatible with my argument that says some real numbers require an infinite description.

Answer 1

But how do you know that there are uncountably many reals? Skolem's paradox is addressing exactly that. If $M$ is a countable model of $\sf ZFC$, then $M$ contains countably many reals. Yet internally to that model, the reals are uncountable.

The math tea argument is, when presented correctly, the following:

Let $\cal L$ be the language of analysis, i.e. $0,1,+,\cdot,\exp,\int,\sin,\cos$ and the function symbols for a few other things we want, perhaps. Then in the "natural interpretation" of $\cal L$ in $\Bbb R$ (which is even second, or third order logic) there are only countably many real numbers that are definable, and therefore most reals are undefinable.

But if we start adding more things to the universe, more real numbers can turn up definable. For example, if $r$ is any real number that wasn't definable originally, we can add it as a constant symbol. Or maybe we added a subset $A\subseteq\Bbb R$ such that $A\cap [n,n+1)$ has exactly the number of elements as the $n$th decimal expansion of $\Bbb R$. This allows us to define $r$, and that's just one example.

But all of this happens inside a universe of a set theory, that is much larger and much more complex, so there's a lot more than can be defined there.

So the problem, really, is that "finite description" doesn't tell us what you are allowed to use for the description. And when you start allowing the description to come from the entire mathematical universe, you invariably start having more things that you can describe. Moreover, since the definitions given over the whole mathematical universe are not internal to the universe, they usual arguments simply break down.

Answer 2

I read a book by Y. Manin where he presents a very similar argument for why there exist uncomputable functions. Then he gives a critique of that argument, which I will try to apply here:

We don't know what a $\bf description$ formally is. This may not seem to be a problem, since for any reasonable way to formalize the notion it is going to be a finite sequence of letters under a finite alphabet. But can we assume that we can work with a single such formalization? Maybe for any real number, we can find an intuitively valid description of it, but there is no formalization that can capture them all.

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Externally, it is possible (but not necessary) that every real number can be uniquely expressed in the same metalanguage you use to describe the set theory you're using.

I believe that here $\textit{to be uniquely expressed in a metalanguage}$ means $\textit{to have an intuitively valid description}$.

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Apparently there's a model in ZFC that can specify each set - and thus real number.

I am not very knowledgeable in this area and can be totally wrong, but here is what I think:

Models do not specify anything. Formal theories specify elements of their models. The answer to the question suggests that there is a model of ZFC, each element of which can be specified in ZFC. So we are not talking about specifying all sets in general, but all sets that are elements of that particular model (and there must be countably many of them, I believe).