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6/21/2025, 7:25:53 PM
6/17/2025, 4:45:37 PM
>>16700006
>we should reject all three
And then what? Suddenly all your proofs in the very basics of any discipline become unreasonably complicated just for some “philosophical” reason
>no Hamel basis guaranteed for every vector space
>some free modules are suddenly not free anymore, have to come up with tedious criteria to see what went wrong
>basically any result about lattices (topologies on a set, group lattices, ring ideals, manifold atlases, etc etc) is thrown out the window and you have to develop a whole framework for each to decide if a maximal element exists
>the only area not affected is analysis because Schauder bases of TVS can rely on Dependent Choice instead
>we should reject all three
And then what? Suddenly all your proofs in the very basics of any discipline become unreasonably complicated just for some “philosophical” reason
>no Hamel basis guaranteed for every vector space
>some free modules are suddenly not free anymore, have to come up with tedious criteria to see what went wrong
>basically any result about lattices (topologies on a set, group lattices, ring ideals, manifold atlases, etc etc) is thrown out the window and you have to develop a whole framework for each to decide if a maximal element exists
>the only area not affected is analysis because Schauder bases of TVS can rely on Dependent Choice instead
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