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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Replying to @DRMacIver @Nathani02617995
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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Anyway the easy way to see that the nonstandard reals are not a model of the axioms for the reals is that the natural numbers have an upper bound but no least upper bound in them.
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Replying to @DRMacIver @Nathani02617995
It has been a long time, apparently!
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Replying to @Meaningness @Nathani02617995
No excuse! I've never forgotten the lessons I learned at the same time you were learning this. (toilet training and eating solid food)
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I will be forgetting those soon enough
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Replying to @DRMacIver @Nathani02617995
Having watched both my parents descend into dementia and incontinence, it’s salient
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