From Wikipedia on the busy beaver,
there is a true-but-unprovable sentence of the form "$Σ(10↑↑10) = n$", and there are infinitely many true-but-unprovable sentences of the form "$Σ(10↑↑10) < n$".)
My question is that for any busy beaver function its well defined and only a finite amount of things need be checked. How then can this statement be true?
For intuition: the $n$th Busy Beaver number is basically a bound on the time a sufficiently small halting machine can take.
In order to verify the $n$th Busy Beaver number, we must execute some algorithm to check it. Our algorithm can be executed by some halting machine (indeed, if it doesn't halt, then it's not fit for purpose because it doesn't answer "yes" or "no" to its question). But if $n$ is large enough, our checking-machine falls into the set of machines the Busy Beaver function is talking about, and we get a self-reference: the checking procedure only gets BB(n) steps to verify that it, itself, halts in fewer than BB(n) steps.