I mean not that you actually need the completeness theorem for this - we can construct models of all these things perfectly well without it - but it's funnier if I put it that way.
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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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Unless you're talking about a deeper level of what mathematical objects really are/whether they exist.
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Also that!
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