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@NikWeaver I wouldn't put it that way, but it's somewhere along the lines of what I always thought (I don't have expert knowledge of set theory, so that might very well be far from the truth). How do you judge if an axiom is true if not by its consequences?
@NikWeaver "We can't decide whether an axiom is true based on whether we like its consequences". But isn't that how axioms where introduced? People didn't prove theorems from set-theoretic axioms before the 20th century, and when the axioms were invented, weren't they designed in such a way that they would express all that people thought was true mathematics as true theorems?
I had a quantum mechanics course last semester where they actually defined $L^2$-spaces like that, but still didnt have any problem stating $e^{ikx} \in L^2$. Physicists are weird sometimes.