>>10600741>all methods of determining whether or not you would want the continuum hypothesis to be true are indistinguishable from just taking it as an axiom
There probably is a result like that if you attach some extremely technical adjective to the front of every noun. But no such result can rule out a new and convincing way of thinking of something. Example: not everyone was easily convinced that the lambda calculus modeled all algorithmic computation, because obviously it's weird looking; but when Turing's machine model turned out to be equivalent, it won over 99% of people. This is creative and social process isn't something you can easily formulate a theorem about.>>10601053>why does there have to be a single "canonical" axioms system?>This boomer set theorist idea is totally dated in the age of topos theory and pluralist mathematics.
This point of view is called "formalism" and it is older than boomers, and a lot of boomer-era set theorists are formalists too. But like finitism, it's a tricky position to defend: you have to believe, more or less, that exists but its power set does not. Or that exists, but not its power set. Or that they "exist", but are inscrutable in some strong way: hard even to get your head around.
In any case I think even a non-formalist will still like alternative axiom systems most of the time. Not topoi though, those are top fucking gay