On 18 Jul 2006 09:56:46 -0700, "MoeBlee" <email@example.com> wrote:
>Lester Zick wrote: >> On 17 Jul 2006 17:24:07 -0700, "MoeBlee" <firstname.lastname@example.org> 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. > >I mentioned theorems provable from axioms. The proofs are easy. But >there's no point in trying to convince you otherwise.
Well, Moe, it's not the truth of theorems which concerns me but the truth and demonstration of axioms. If you want to assume geometric objects as auxilliary notions within set theory definitions for circles go ahead. Just don't try to tell me they define anything with set theory.
>> >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. > >So you have some view of set theory based on what "has often been >stated to you".
I have no idea what set theory may be apart from what has been conveyed to me by others. I doubt many do.
> There is an empty set axiom sometimes stated, but it is >superfluous, as it is derivable from the axiom schema of separation and >the axiom of extensionality. As to a suc(0) axiom of set theory, I >still don't know what you have in mind, since you don't.
You might consider asking those who use the phrases themselves unless you yourself intend to claim you invented the subject.
>> >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. > >Seems to you, but not to anyone who can follow a basic mathematical >definition. As I defined a 'circle', it is a set of points in RXR all >equidistant (for a distance greater than 0) from a given point. That's >a circle, not a sphere.
Actually not unless you assume your points all lie on a plane to begin with which is a geometrical assumption and not a set definition.
>> >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. > >There's no lack of ability to express 'dimension', 'space', and >'n-dimensional space' in set theory.
To use the words perhaps not. But the objects referred to are nonetheless geometric not numeric objects. When used in set theory definitions they're nothing but geometric not arithmetic auxilliaries.
>> 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. > >'straight line' and 'plane' are easily defined in set theory.
Sure they are:
"A straight line is the set of all points on a straight line".
"A plane is the set of all points on a plane".
"Universal truth is the set of all points which are universally true."