This is a fair point, but I'm referring to information in the information theoretic sense; in this technical sense, mathematical truths are indeed not information.

There are proofs that rely on the GCH or Large Cardinal Axioms or V=L which are not among the accepted axioms and proven to be independent of the other axioms.

I'm aware that the Axiom of Choice is required for some important results of practical import (Tychonoff's theorem, for example, is equivalent to it), but do you know of any important and useful results following from the GCH, etc.? I've only looked into this a little; foundational math is not really my field.