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on Smallest $n$ for which the value of $BB(n)$ is undecidable ($C_{14}$)

The statement

$C_{14}$ is the smallest $n$ , such that the value of the busy beaver number $BB(n)$ is undecidable in ZFC (or equivalently ZF). Explicitly, it is the smallest $n$ such that there is a Turing machine with $n$ states for which it cannot be proven in ZFC (assuming ZFC is consistent) whether it halts or not.

The existence of $C_{14}$ essentially follows from Gödel's second incompleteness theorem and the fact that Turing machines are strong enough to encode ZF.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.