- “interactive game” = physicist or mathematician attacking a theory with observation or proof sequence
- “Adversary” = the physical or math theory.
- “Challenger” = physicist or mathematician.
- “Efficient challenger” = Godel’s “effective procedure” which can be a computer or person writing on paper, as long as it does not go on to infinity
I can’t see how this definition of falsifiability in crypto is not subject to Godel’s theorem.
From wiki on Godel’s 1st incompleteness theorem:
completeness = “A set of axioms is complete if, for any statement in the axioms’ language, that statement or its negation is provable from the axioms”. Godel’s theorem is this does not exist if the axioms are finite and consistent.
“Zcash is protected against crack X” where X has been expressed in Zcash’s finite, consistent axioms. Godel’s theorem says there exists some X for which the assertion can’t be proven.
Therefore, by Godel’s theorem, there is no crypto made up of mathematically-consistent axioms for which it can be proven all cracks are impossible. You would have to know all possible cracks and then work backwards to get your crypto axioms which would not be consistent. All possible cracks can’t be known. But for a finite range of cracks, the axioms could be consistent.
[ edit: I guess that’s the difference: crypto works backwards from the known cracks, so its “axioms” do not need to be mathematically consistent, or for a limited range of cracks, the axioms could be consistent. Godel’s 2nd theorem may apply better. That is, in order to defend against a wide variety of attacks, 2 or more inconsistent axioms have to be used. I’m thinking about husband-wife teams and the principle of “complimentarity” in physics: the very nature of trying to disprove a wide variety of statements (making true and consistent observations) creates a inconsistent axioms if they are to be few in number. ]