On 17 Jul 2006 17:24:07 -0700, "MoeBlee" <email@example.com> wrote:
>Lester Zick wrote: >> Technically set theory only assumes the set of real numbers by >> assuming a natural number and the suc( ) axiom. > >In Z set theory, from just the axiom schema of separation and the axiom >of extensionality, we prove the existence of a unique empty set, which >then fits the definition of the least natural number. And for any given >natural number, we can prove its existence, even without the axiom of >infinity. And we prove the existence of the set of natural numbers from >the axioms but now including the axiom of infinity. And then the >existence of the set of real numbers is proven (and we can construct >different structures that are complete ordered fields, all of which are >isomorphic).
Well, Moe, it's good to see you have found nothing better to do with your time than to restate vacuous claims amounting to nothing more than what you assume to be true.
>So I don't know what assumption of the existence of a (particular) >natural number you have in mind and then what axiom you mean by the >"suc( )" axiom.
It has often been stated to me as zero and suc(0). I don't agree that zero is a natural number but for what it's worth there it is.
>As to circles, we can define them in set theory: > >Let X stand for Cartesian product (which is definable in set theory). >Let R stand for the set of real numbers (which is definable in set >theory). >Let d stand for the standard distance function on RxR (which is >definable in set theory). >Let > stand for the standard ordering on R (which is definiable in set >theory). >Let 0 stand for the real number zero (which is definable in set >theory). > >c is a circle <-> EpEe(p in RXR & e>0 & Az(zec -> (z in RXR & d(p z) = >e))) > >I.e., a circle c is a set of points in RXR such that there is a given >point p and a given non-zero distance e such that each member of c is e >distance from p.
Except this seems to define a sphere. Perhaps Bob Kolker can give you a hand in this respect if not in defining a circle certainly in construting pointless definitions which you assume to define a circle.
>And this definition can be adjusted to define 'is a circle' in other >n-dimensional spaces.
Except we don't quite have a definition for a dimension in set theory to begin with. Of course it helps if we can just assume we know what we mean by straight lines, planes, so forth and so on but it's a little difficult to see what we mean by such assumptions without set definitions for them.