On 19 Jul 2006 11:15:56 -0700, "MoeBlee" <firstname.lastname@example.org> wrote:
>Lester Zick wrote: >> On 18 Jul 2006 15:21:01 -0700, "MoeBlee" <email@example.com> 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.
Yeah too bad an accurate definition for a circle isn't met as well.
>> >> 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.
In other words I have a view provided by others as do you.
>> >> 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.
Sure like definitions of spheres you choose to call circles.
>> >> >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.
In RxR? Is that perchance another name for where points fall in Spain?
>> >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.
Oh if I could just assume whatever I wanted I could sure as hell draw the prettiest definitions of circles in set theory you ever saw. Just wouldn't make them circles.
>> 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'.
Which turned out to be nothing more or less than a sphere intersected by an undefined geometrical object also known in the parlance as a plane which you just assumed without definition.
>> >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'.
For which you just assumed a plane.
>> 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.
So now a definition becomes a definitional axiom in which you just assume geometric objects like planes without definition?