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army1987 comments on Standard and Nonstandard Numbers - Less Wrong
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Comments (83)
I don't see yet how this connects to the other posts from the epistemology sequence, but it's definitely nice. I've wanted to learn more mathematical logic for some time. I didn't quite understand why exactly using an axiom schema isn't as good as using second order logic, before I read this post.
I read that "any" as "for at least one", rather than as "for every". That confused me quite a bit. Maybe native speakers won't have a problem with that, but to me the connection between "any" and "some" is too close.
It's also not clear to my where the order relation comes from.
Probably it's because of the “no group” before it; cf “I can do anything” and “I can't do anything”. Negations and quantifiers in English sometimes interact in weird ways, making it non-trivial to get the semantics from the syntax.
Wiktionary gives the meanings "at least one" and "no matter what kind". The first likely doesn't apply here, as it's not used in a negation or question. To interpret "no matter what kind" to mean "every" seems like a stretch to me. I really do think the meaning of "any" is ambiguous here. "any" just specifies that we don't have any further constraints on x. You could replace it with "every" or "at least one", but not with "every even" or "at least one even", as that would introduce a new constraint.
It doesn't, but I was hypothesizing that the reason why on the first read it sounded to you as though it did was the negation (“no group”) before it.