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Topic: Eliminating Quantifiers For Dummies! A(x) E(y) ALL(t)EXIST(u)EaAb ..
Replies: 88   Last Post: Jan 13, 2012 8:47 PM

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 bogus address Posts: 227 Registered: 12/12/04
Re: Eliminating Quantifiers For Dummies! A(x) E(y) ALL(t)EXIST(u)EaAb ..
Posted: Jan 13, 2012 9:58 AM

> Do you want to derive real analysis from first principles or not? It
> would be nice to do it in some stripped-down, so-called "first order"
> system, but you and the experts here agree that it can't be done. So,
> you have to use another system with more "features" if want to do real
> analysis, right?

If you want to use the techniques of non-standard analysis to get
slicker proofs, you state the problem in a restricted first-order
formalism and use more powerful systems at the semantic level, so
you can appeal to elementary equivalence. You have to use both at
once.

It's a commonplace of logic that you trade off semantic expressiveness
against deductive tractability. Fragments of first-order group theory
can be very efficiently mechanized using rewrite rule techniques. No
such techniques can ever work for set theory. I'd be surprised if the
challenge problem I set you hasn't already been solved by a mechanized
proof search in some term rewriting system, with no human intervention
required. Meanwhile it looks like your system can't even *express*
the problem, so mechanically checking an interactively generated proof
of it will be impossible.

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