Are there mathematical objects that have been proved to exist but cannot be described in words?

You'd have to be more precise about what you mean by "describable in words". But you could argue that some irrational numbers can be described in words, like $\pi$ is the ratio of the circumference to the diameter of a circle. So if you consider each real number a mathematical object, then we could never possibly describe all of them because the set of things we can describe in language is necessarily countable, there are only a countable number of ways to assemble letters into words into sentences. But there are uncountably many real numbers. So inevitably most of them could never be described.

The reals are describable as a set, but you cannot describe each and every one of them in a way that distinguishes their individuality. Incidentally we can describe each and every element of $\mathbb N$ individually, because every natural number can be represented by a unique finite sequence of characters in the set $\{0,1,\dots,9\}$. The same cannot be said of $\mathbb R$.

Anything coming from the axiom of choice really. Such as the well ordering of the reals. Is there a known well ordering of the reals?

We can do it using set theory. The number of definable objects is countable, but the number of things that exist is uncountable. So something exists which isn't definable.

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Interestingly, you can't express definability in ZFC, in ZFC. If you try to, you get the following contradiction: https://mathoverflow.net/a/204794/1682.

To give another example from non-standard analysis (i.e. analysis with infinitesimals):

In non-standard analysis you extend the set of reals in a similar way you extend $\mathbb Q$ to $\mathbb R$ with Cauchy sequences:

1. You take the set of all real sequences.
2. You define a equivalence relation on the set of sequences. Two sequences $(x_n)_{n\in\mathbb N}$ and $(y_n)_{n\in\mathbb N}$ are equivalent if $x_n=y_n$ "for almost all $n\in\mathbb N$".

To define the concept "for almost all $n\in\mathbb N$" so called ultra filter are used, which existence can be shown with the axiom of choice. Until now no constructable proof is known. So you can show that

• The equivalence class of $(\tfrac 1n)_{n\in\mathbb N}$ will be an infinitesimal, i.e. a positive nonzero number smaller than each $q\in\mathbb Q^{+}$.
• The equivalence class of $(n)_{n\in\mathbb N}$ will be infinitely big.

But you will not know, whether the equivalence class of the sequence $(0,1,0,1,0,1,0,1,\ldots)$ represents 0 or 1.

(Note: What $(0,1,0,1,0,1,0,1,\ldots)$ is, depends on the taken ultra filter. There are some, where this sequence represents 0 and some where the sequence represents 1).

Things that are not describable in words exist as disturbances, obstacles to our understanding of the world, until we make them describable, i.e. until we find a conceptual metaphor rich enough to bring them to existence in language, as well as in conception. This is the adventure of science. In the history of mathematics this is particularly evident. There was once a time when irrational numbers weren't "describable in words". Now they are taught to children.

Are there objects that can keep resisting our attempts to tame them? I think there are, and some of the other answers tried to explore some of these "occurrences". More than once a mathematician was able to demonstrate things s/he couldn't convince him/herself of. It can be argued that these objects are part of our ordinary experience of the world, and we just don't mind their presence. This ineffable dimension of being is what some philosophers call the Event, or the Real.

Some say that modern science has recently found unsurpassed limits in its ability to digest the abnormalities it encounters, limits it can no longer expect to negotiate. And yet, it keeps describing new things. In words, of course. And also designating them by symbols, or creating images of them.

Will there always be objects that are indescribable in words, waiting to be "domesticated"? Now, that is a very interesting philosophical problem.