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Topic: FLT Discussion: Simplifying
Replies: 65   Last Post: Mar 17, 2001 11:59 PM

 Messages: [ Previous | Next ]
 oooF Posts: 96 Registered: 12/13/04
Re: FLT Discussion: Simplifying
Posted: Jan 21, 2001 9:36 PM

"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news://G7JJ6G.F4C@cwi.nl...
> In article <A17a6.1877\$TI3.5836@nntpserver.swip.net> "oooF"
<fooo@swipnet.se> writes:
> > "Randy Poe" <randyp@visionplace.com> wrote in message
> > news://3a67971c.438991574@news.newsguy.com...
> >
> > [...]
> >

> > > Then perhaps you can be made to understand that mathematical theorems
> > > are built up from a few small starting axioms. A theorem is built on
> > > results you have already established.

> >
> > I told JSH (perhaps 6 months ago in a thread here during the

'tautological
> > proof'-period) about how 'everything' is derived from a set of axioms
that
> > are considered true (they are true because we say they are true).
>
> In a sense only when we do algebra. In algebra the axioms (like a+b =

b+a)
> are more like properties. You have to prove that your operators (+ and *)
> and the set of elements for which you define them indeed *do* satisfy the
> axiom, or rather, have the property you wish. It is (in my opinion) a
> misnomer to call these things "axioms". I see the following "axioms"
> as standard:
> R1: a + (b + c) = (a + b) + c
> R2: a + b = b + a
> R3: there is a 0 such that a + 0 = 0 + a = a
> R4: there is a -a such that a + (-a) = (-a) + a = 0 (this implies R3)
> R5: a * (b + c) = a * b + a * c and
> (a + b) * c = a * c + b * c (as Keith Ramsay correctly observed)
> R6: a * (b * c) = (a * b) * c
> R7: a * b = b + a
> R8: there is a 1 such that a * 1 = 1 * a = a
> D : a * b = 0 implies a = 0 or b = 0 or both (this implies R3)
> F : there is a a^(-1) such that a * a^(-1) = a^(-1) * a = 1 (implies R8).
>
> So when you come up with a set of "elements" and with operations + and *
> on them you have to show what of the properties R1 to R8, D and F are
> satisfied, and only when you have done this you can assume results for
> those kind of things. Now whether a field implies R7 or not depends
> simply on nomenclature (yes, when I did this stuff I learned both).
> But when we talk about properties of integral domains (something like
> R1 to R8 + D) we have first to prove that the operations satisfy the
> required properties.

I agree with you (exept R7 above), and would like to see JSH define (+, *)
and verify his 'pattern ring' R against R1-R6 and also that R is closed
under both operations.

Date Subject Author
1/15/01 jstevh@my-deja.com
1/15/01 Dik T. Winter
1/16/01 Charles H. Giffen
1/16/01 jstevh@my-deja.com
1/16/01 Randy Poe
1/18/01 jstevh@my-deja.com
1/18/01 Michael Hochster
1/18/01 Peter Johnston
1/18/01 Randy Poe
1/18/01 Doug Norris
1/16/01 Doug Norris
1/16/01 Randy Poe
1/16/01 Dik T. Winter
1/18/01 jstevh@my-deja.com
1/19/01 Dik T. Winter
1/19/01 Randy Poe
1/20/01 jstevh@my-deja.com
1/20/01 oooF
1/21/01 hale@mailhost.tcs.tulane.edu
1/21/01 Peter Percival
1/21/01 Randy Poe
1/26/01 Franz Fritsche
1/19/01 gus gassmann
1/20/01 jstevh@my-deja.com
1/20/01 Doug Norris
1/26/01 Franz Fritsche
1/16/01 hale@mailhost.tcs.tulane.edu
1/16/01 Randy Poe
1/17/01 hale@mailhost.tcs.tulane.edu
1/18/01 jstevh@my-deja.com
1/19/01 hale@mailhost.tcs.tulane.edu
1/20/01 jstevh@my-deja.com
1/21/01 hale@mailhost.tcs.tulane.edu
1/18/01 Peter Percival
1/19/01 hale@mailhost.tcs.tulane.edu
3/17/01 Ross A. Finlayson
1/16/01 hale@mailhost.tcs.tulane.edu
1/18/01 jstevh@my-deja.com
1/19/01 hale@mailhost.tcs.tulane.edu
1/29/01 jstevh@my-deja.com
1/19/01 Dik T. Winter
1/21/01 Dennis Eriksson
1/15/01 Michael Hochster
1/16/01 jstevh@my-deja.com
1/16/01 Michael Hochster
1/18/01 jstevh@my-deja.com
1/18/01 Peter Percival
1/18/01 Randy Poe
1/19/01 oooF
1/21/01 Dik T. Winter
1/21/01 oooF
1/18/01 Edward Carter
1/19/01 W. Dale Hall
1/19/01 Michael Hochster
1/16/01 Randy Poe
1/16/01 Randy Poe
1/17/01 W. Dale Hall
1/17/01 W. Dale Hall
1/19/01 oooF
1/16/01 Charles H. Giffen
1/16/01 David Bernier
1/16/01 jstevh@my-deja.com
1/18/01 Arthur
1/30/01 plofap@my-deja.com
1/30/01 plofap@my-deja.com
1/30/01 plofap@my-deja.com