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?
Under certain reasonable assumptions, the independent statements are of full density. See https://www.cs.auckland.ac.nz/~cristian/aam.pdf
Edit: As David E Speyer points out, the results of the above paper are, to say the least, questionable. Thus, all we can say is that the proportions of true, false, and independent statements are all positive. Precisely what these proportions are would depend on how we code the statements and measure their quantity.