|
|
Re: Eliminating Quantifiers For Dummies! A(x) E(y) ALL(t)EXIST(u)EaAb ..
Posted:
Jan 11, 2012 6:14 PM
|
|
On Jan 11, 5:47 pm, Jack Campin <bo...@purr.demon.co.uk> wrote:
> >>> Just to be clear, do you see the axioms of first-order group theory
> >>> (Mendelson's 7 axioms with their explicitly unbounded quantifiers)
> >>> to be a workable, alternative set of axioms for that branch of
> >>> mathematics we call group theory, or not.
> >> Just to be clear, do you see the axioms of Peano arithmetic to be a
> >> workable, alternative set of axioms for that branch of mathematics
> >> we call real analysis, or not?
> > Haven't you heard? The real numbers can be constructed from the set of
> > natural numbers. I have made a modest start to toward that end myself
> > using my program, proving the irrationality of the square root of 2
> > (previously posted here IRRC).
>
> 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.
> 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).
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Also see "The Barber Paradox Video"
|
|
