According to Skolem's Paradox, ZFC as a first order axiomatization of set theory has a countable model, but allows a proof that uncountable sets exist in every model of ZFC.
It becomes counter-intuitive if we accept ZFC as modelling our reality, a formal model of what we think a set is, but also accepting Cantor's proof that $\mathbb R$ is uncountable.
The paradox could be resolved by looking at the notion of "uncountability" within the axiom system (and applied to a particular model), and externally (a notion we have of uncountable outside of the axiom system). This is the explanation from nlab, to cite:
The resolution of this apparent paradox is that, while this conclusion is true internally, it is not true externally: namely any two infinite sets are countable externally in that model, hence there is a $1$–$1$ function between any two of them including for a model of some uncountable set $X$ and of its power set $P(X)$. However, that function (or its graph) is not in the model! One can enlarge the model by adding the function (and more). But this extended model will necessary have $P(X)$ uncountable externally and there is no $1$–$1$ function from $X$ to $P(X)$ externally any more.
Now assume that ZFC captures what we think of as a set. The real numbers seem perfectly uncountable for us, and somehow they capture the idea of approximation on the number line. Would it be possible that another observer with "more insight" looks at us, at our model of ZFC, and for them this would look countable, as they are able "externally" to put our construction of $\mathbb R$ in correspondence from our construction of $\mathbb N$.
I would say no, simply because I cannot imagine, simply because I think I believe in the construction. But Skolem says that we cannot be sure about our interpretation of ZFC. But what do you think, may we be trapped in some model of thinking? Is this a conceivable scenario, another observer being smarter than us, seeing our reasoning "externally"? But then this seems to be a new incarnation of Skolem's paradox right from its resolution, with giving us some sort of epistemological boundary...