Lester Zick wrote: > 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.
I have no interest in convincing you of the benefits of set theory. But in stating my own observations, in contrast with yours, I note that set theory does allow precise definitions of such things as 'is a circle'.
> I have no idea what set theory may be apart from what has been > conveyed to me by others. I doubt many do.
I know what set theory is, as it is defined in certain systematic treatments of the subject.
> 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.
I don't know of any treatment of set theory that has axioms named 'suc( )' or 'suc(0)', so since you don't know what they mean, I can only let them rest as meaningless verbiage.
> >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.
I corrected to say that a circle in RXR is a sphere. As to where points lie, geometry, and set theory, you don't have to accept this for yourself, but it is easy to express basic geometric principles in set theory.
> >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.
Whatever you call the abstractions, whether 'geometric', 'numeric', or 'arithmetic auxillaries', they are formalizable in set theory.
> >'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." > > See how easy that was?
By 'defined', obviously, I don't include circular definitions.