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Re: FLT Discussion: Simplifying
Posted:
Jan 21, 2001 8:37 PM
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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.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
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