Lester Zick wrote: > On 19 Jul 2006 11:15:56 -0700, "MoeBlee" <email@example.com> wrote: > > >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. > > 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?
A good beginning discussion of the subject of mathematical definitions is in Suppes's 'Introduction To Logic'. But in order not to inhibit the metastasis of your own convictions, I recommmend that you not read such books.