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Re: Eliminating Quantifiers For Dummies! A(x) E(y) ALL(t)EXIST(u)EaAb ..
Posted:
Jan 11, 2012 7:58 PM
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On Jan 11, 6:47 pm, Jack Campin <bo...@purr.demon.co.uk> wrote:
> >> The parallel is: your attitude to limited formalisms like FO group
> >> theory is like saying we don't need integers because everything we
> >> might want to measure or calculate can be measured or calculated
> >> with real numbers.
> > I don't see the parallel. As I understand it, regular group theory
> > is not derivable from the axioms of first-order group theory. Real
> > analysis, however, CAN be derived from Peano's axioms.
>
> It can't.
I believe you are mistaken.
> You need set theory as well (or something comparable, like
> type theory or higher order logic). Mendelson defines PA for you,
> look it up.
>
I don't think so. Here is my version of the Peano axioms:
1 Set(n)
2 1 in n
3 ALL(a):[a in n => next(a) in n]
4 ALL(a):[a in n => ~next(a)=1]
5 ALL(a):ALL(b):[a in n & b in n & next(a)=next(b) => a=b]
6 ALL(a):[Set(a) & 1 in a & ALL(b):[b in n & b in a => next(b) in a]
=> ALL(b):[b in n => b in a]]
> >> Integer arithmetic serves a mathematical purpose despite being less
> >> expressive than real analysis.
> >> First order theories like that of group theory serve a mathematical
> >> purpose despite not being a lot of use as a general foundation for
> >> mathematics.
> > Sorry, I just find it hard to take seriously a version of group theory
> > stripped of any references to an underlying set, if that is indeed the
> > case (I still have some doubts about that, but have passed caring).
>
> You have been given an example of how the theory can be used (the proof
> of the Ax-Kochen theorem).
The point is, any theorem that can be derived in the FO version of
group theory can supposedly be derived in the regular version of group
theory. I will stick to the regular version, thanks.
> And if your system can't prove theorems in it, it's a flop.
First order group theory seems to be a pretty obscure little specialty
by the looks of it. Besides, there is plenty of mathematics based on
an underlying set. Number theory, abstract algebra, analysis and
geometry come to mind.
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Also see "The Barber Paradox Video"
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