> You can take any bitstring (but not all bitstrings) of any length, including infinite, and assign a specific shorter (and finite) bitstring to represent it. Y

> You can take any bitstring (but not all bitstrings) of any length, including infinite, and assign a specific shorter (and finite) bitstring to represent it. You then have a finite encoding for that specific infinite sequence of symbols which can henceforth be used to describe it. You just can't do it for every infinite bitstring because you have insufficiently many finite bitstrings with which to represent them. I agree with your general idea, but, won't you run into a problem already before you run into the pigeon-hole principle? How do you do that, even for a single one? I mean, how do you "take" an infinite bitstring to assign a symbol to it, without first having to have a finite description of this infinite bitstring? You can't say "this infinite bitstring is now represented by the symbol 01101", without directly or indirectly specifying what "this infinite bitstring" refers to, and you need to do that specifying in a finite amount of symbols. So it seems to me that, if you were to (foolishly) try the above infinite task, you'd run out of infinite bitstrings (that you can describe) just as fast as you'd run out of finite bitstrings that you'd like to compress them with?