On 18 Jul 2006 15:21:01 -0700, "MoeBlee" <firstname.lastname@example.org> wrote:
>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'.
If by "precise definition" you mean "arithmetic assumption" I agree.
>> 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.
Which you divined all by your lonesome and was not conveyed to you by others?
>> 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.
Alas I have been lead astray once more!
>> >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.
Good. So when asked for the set theory definition of a circle you provided the set theory definition of a sphere. It's good to know you understand set theory well enough to provide incorrect definitions at the drop of a hat.
> 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.
But it isn't quite so easy to provide correct definitions for geometric objects in set theory apparently. All you can do is express what geometric objects sets of all points describe according to what geometric figures they conform to. That's not a definition. It's a description.
>> >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.
Yes well we're still waiting for these formal definitions of a circle 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.
And by "defined" apparently you don't include set definitions either.