This is a kind of soft question but I prefer to ask it here where people understand more mathematics rather than on some philosophy or language forum.
An instance of Cantor's theorem, the Grelling paradox gives rise to the following.
Consider the set $Adj$ of all (English) adjectives. We have the following function $f : Adj × Adj \to \{0,1\}$ defined for all adjectives $a_1$ and $a_2$,
$$f(a_1,a_2)=\cases{1 ~~\text{ if } a_2 ~\text{ describes } ~a_1; \\0 ~~\text{ if } a_2 ~\text{ does not describe } ~a_1.}$$ so with $\neg:\{0,1\}\to \{0,1\}$ which sends $0$ to $1$, and $1$ to $0$ we arrive at the following diagram $\require{AMScd}$ \begin{CD} Adj\times Adj @>f>> \{0,1\}\\ @A \Delta A A= @VV \neg V\\ Adj @>>\chi> \{0,1\} \end{CD} where $\chi$ characterize some subset of adjectives that can not be described by any adjectives (by Cantor's theorem).
I wonder, if this (or some similar) construction tells us that there is a concept which has no definition in English (a natural language)?
EDIT: At first sight the term "concept" in the initial variant of question is rather ambiguous; in a vide sense a concept for me means everything we percept by consciousness.
On the other hand I feel comfortable to think that all naive-set theoretic collections are kinds of concepts. But inverse is not clear for me, I like to think that there may be concept which does not form a collection. In order to eliminate ambiguity of the term "concept" readuce a search area and let us consider only members of $\mathcal P(\mathbb N)$ which is uncountable.
Then as lulu suggest there are uncountably many concepts and by Cantor's theorem, counting argument shows that some subsets of $\mathbb N$ lack of definition in English.
And as I see a situation doesn't change if instead of English I consider some countably infinite language. And definition is defined as finite collection of words in the language.
Sorry if set-theory tag is not relevant at least it may attract attention of the people who have thought about something similar.