In article <A17a6.1877$TI3.firstname.lastname@example.org> "oooF" <email@example.com> writes: > "Randy Poe" <firstname.lastname@example.org> wrote in message > news://email@example.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/