Let us enumerate all statements of PA or ZFC by length, upto n characters, then in the limit as $n\rightarrow\infty$, what proportion of statements are provably true, provably false, or independent?
Ok, no enumeration necessary, just count all of length less then n and take the limit.
Is it perhaps 50%,50%,0%?
What if we discard all statements which are simply the negation character in front of a shorter statement?
What is the asymptotic density of independent statements?
Is any non-trivial results of this sort known for any theory?