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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Replying to @DRMacIver @Nathani02617995
I will be forgetting those soon enough
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Having watched both my parents descend into dementia and incontinence, it’s salient
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