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Skatche comments on The benefits of madness: A positive account of arationality - Less Wrong
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This is a false assertion, they are only true if the axioms used to conclude them correspond to reality. There are proofs that rely on the Axiom of Choice which is not accepted (as far as I can tell) by everyone on this site (as well as the axiom of infinity?). 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.
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.
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.
Game theoretic results that are generalized to infinite games often require the use of the GCH. For instance see "Variations on a Game" by J Beck 1981.