The axiom of choice seems trivial, yet it gives rise to Banach Tarski paradox. Monkey confused.
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@St_Rev you have inspired me to get out my oxford encyclopedia of mathematics and start poking at it again.Thanks. Twitter will use this to make your timeline better. UndoUndo
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@MorlockP IOW, if you think the AoC is trivial, go make an infinite number of choices and let us know how that worked out for you. -
@St_Rev so there's something I'm missing here re AoC. Is it the complexity of defining how to choose on arbitrary inputs? -
@MorlockP AoC means you can choose even when you have no knowledge about the sets. Sets are minimally structured, they're just bags. -
@MorlockP You don't need AoC to make infinity choices if you can supply a rule. Like, if you have a zillion copies of the integers... -
@MorlockP ...you don't need AoC to say 'ok I'll set n_i = i'. But an infinite number of choices *with no rule for choosing* requires AoC. -
@MorlockP Specifically in Banach-Tarski, AoC sneaks in when you need a set of coset representatives of the funky tree subgroup thing.
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@BenignWit@MorlockP Yeah, it's bread and butter in most areas. Constructing the real numbers and defining continuity in particular need it. -
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@BenignWit@MorlockP PhD in math by training. Disabled shut-in by trade. -
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@BenignWit@MorlockP Me too.
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. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.