Also topology should be taught as part of a first course on set theory and logic and real analysis should be taught assuming it as a foundation.
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Replying to @DRMacIver
That would have saved me a year’s worth of existential crisis when I was 19
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Replying to @DRMacIver
I was existentially incapable of continuing with math until I could find out what a real number was, which my analysis teacher refused to tell me. After six months of clinical depression I found the answer in the library and life became livable again.
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Replying to @Nathani02617995 @DRMacIver
Well the answer I *got*, which seemed completely satisfactory at the time, was “Dedekind cuts/Cauchy sequences.” This turns out to be wrong, but I didn’t figure that out until MUCH later, >
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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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Replying to @Meaningness @Nathani02617995
"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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Replying to @DRMacIver @Nathani02617995
Wait I haven’t done this stuff in 35 years, but the nonstandard reals are a model for the reals axioms, and not isomorphic, right? Or am i misremembering?
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Replying to @Meaningness @Nathani02617995
The nonstandard reals are not a model for the real axioms - they don't satisfy the completeness property. They satisfy all the same first-order sentences, but the real axioms involve quantifying over sets of reals, which is not a first order sentence.
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Ugh, it’s been a long time… if you go beyond fopc, you’ve got a ton of big new philosophical problems if my memory isn’t completely rotten
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Replying to @Meaningness @Nathani02617995
Well it's still first order predicate calculus, it's just in the theory of sets rather than the theory of ordered fields.
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(Sorry, I should have been clearer about what I meant about satisfying the same first order sentences, but I'm super rusty here too)
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