Lester Zick wrote: > On 18 Jul 2006 15:21:01 -0700, "MoeBlee" <firstname.lastname@example.org> wrote: > >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.
No, I don't mean 'arithmetic assumption'. I mean a definitional axiom, which entails that the criteria of eliminability and non-creativity are met.
> >> 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?
Conveyed by my reading of books, and by my compiling results and proofs from different books into one ongoing document of my own, and by the help of other people giving added explanations. That is opposed to having only a haphazard view of basic set theory.
> >> 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!
You get better view from reading textbooks.
> >> >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.
No my formulation of a definition of 'is a circle' is fine; what I had to correct is my error in saying that a circle in RXR is not also 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. > > But it isn't quite so easy to provide correct definitions for > geometric objects in set theory apparently.
I'm not aware of any basic geometry that can't be formalized in set theory. I'm confident that if you knew basic predicate calculus and basic set theory, then you'd be able to make such formulations yourself.
> 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.
I gave a fine definition, in biconditional form, of the predicate 'is a circle'.
> >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.
I already gave a definition of 'is a circle'.
> And by "defined" apparently you don't include set definitions either.
By 'defined', in this context, I mean having given a definitional axiom, which entails that the criteria of eliminability and non-creativity are met.