Higher order Quines - when do super Quines exist?
There's a good chance this question is independent of ZFC for most programming languages.  In particular, I'd expect it to be true of any language where the shortest Quine is longer than the Kolmogorov complexity decidability bound, the integer $n$ such that no string can be proven to have Kolmogorov complexity $&gt; n$.  I don't know to prove this, since even if you could express a Quine by a shorter program it wouldn't be a Quine, but I would expect a similar result to apply.

However, if the above argument were true, the argument itself might also be unprovable in ZFC, since it would require showing that no Quines exist below the complexity bound.