The fact that people get hung up on the truth or falsehood of set theoretic axioms like Choice or CH is a weird messy holdout of mathematical platonism and we should be taking Godel's completeness theorem more seriously. (No, not his incompleteness theorem. The other one)https://twitter.com/nex3/status/1199086783222255616 …
and by then I’d developed a meta-rational view so I didn’t have to freak out when rationality broke down.
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"The object satisfying these axioms (which we prove exists and is unique up to isomorphism), pick your favourite implementation" is sufficiently standard fare literally everywhere in mathematics that I'm not sure it's reasonable to count it as a breakdown of rationality.
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Unless you're talking about a deeper level of what mathematical objects really are/whether they exist.
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