This is a post I wanted to write some time ago; I’ve forgotten why, but it was short and cute enough to finish. Our starting point is the following observation:
Theorem 1: Universal lossless compression is impossible. That is, there is no function which takes as input finite strings (over some fixed alphabet) and always produces as output shorter finite strings (over the same alphabet) in such a way that the latter is recoverable from the former.
Annoying Precision 