The Definition of Points
in [Math]
|
Prev: Guide to presenting Lemma, Theorems and Definitions
Next: Density of the set of all zeroes of a function with givenproperties
From: Tony Orlow on 31 Mar 2007 19:12 Virgil wrote: > In article <460e571f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: >>> On 30 Mar, 18:25, Tony Orlow <t...(a)lightlink.com> wrote: >> ~v~~ >>>> An actually infinite sequence is one where there exist two elements, one >>>> of which is an infinite number of elements beyond the other. >>>> >>>> 01oo >>> Under what definition of sequence? >>> >>> -- >>> mike. >>> >> A set where each element has a well defined unique successor within the >> set. Good enough? > > No! if we define the successor of x as x + 1, as we do for the ntaurals, > then the set of rationals and the set of reals, with their usual > arithmetic, both satisfy your definition of sequence. > > A sequence should at least be well ordered and have only one member, its > first, without a predecessor. I agree that what are considered normal sequences have first elements, but I don't see that the integers, or the adics, aren't a sequence in a broader sense, if we choose any arbitrary starting point. We can say a sequence has some single element without predecessor, or some element without successor, or both, so as to limit the line to a ray or segment. But the most general rule is that it may go one forever in both directions, as y -> Ex Ez : x<y<z, ala Archimedes. Yesno? :D Tony
From: Virgil on 31 Mar 2007 19:14 In article <460ee056(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > > If all you mean by "actually infinite" is "uncountable", then > > just say "uncountable". Of course an "uncountable sequence" > > is a contradiction, so you still have to define what you mean > > by a "sequence". > > > > > > Please do expliculate what the contradiction is in an uncountable > sequence. What is true and false as a result of that concept? A mathematical sequence is a function with the naturals as domain. If TO wishes to refer to something which is not such a function, he should not refer to it as a sequence if he wishes to be understood in sci.math. > > > I know you are incapable of actually thinking about all the elements of N, > > but that is your problem. In any case, N is not an element of N. > > Citing Ross as support is practically an admission that you are wrong. > > > > Stephen > > > > Sure, of course, agreeing with someone who disagrees with you makes me > wrong. I'll keep that in mind. Thanks.. It is not so much that Ross disagrees with one person, it is that he disagrees with everyone, frequently including himself.
From: Tony Orlow on 31 Mar 2007 19:15 Lester Zick wrote: > On Fri, 30 Mar 2007 12:07:44 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>>> Those aren't geometrical expressions of addition, but iterative >>>>>> operations expressed linguistically. > >>>>> Which means what exactly, that they aren't arithmetic axioms forming >>>>> the foundation of modern math? The whole problem is that they don't >>>>> produce straight lines or colinear straight line segments as claimed. >>>> Uh, yeah, 'cause they're not expressed gemoetrically. >>> Well yes. However until you can show geometric expression are point >>> discontinuous I don't see much chance geometric expression will help >>> your case any. >>> >>> ~v~~ >> What does point discontinuity in geometry have to do with anything I've >> said? > > You talk about lines as if they were made up of points. > > ~v~~ I do, and the thread is picking up. And, that's not why. :) Or, maybe it is. No point lies independent of any space, or it's insignificant. No point is defined except as different in however many directions are under consideration. Where points are so defined, they allow for lines. :) 01oo
From: Tony Orlow on 31 Mar 2007 19:15 Lester Zick wrote: > On Fri, 30 Mar 2007 12:08:06 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>>> So, start with the straight line: > >>>>> How? By assumption? As far as I know the only way to produce straight >>>>> lines is through Newton's method of drawing tangents to curves. That >>>>> means we start with curves and derivatives not straight lines.And that >>>>> means we start with curved surfaces and intersections between them. >>>>> >>>> Take long string and tie to two sticks, tight. >>> Which doesn't produce straight line segments. >>> >>> ~v~~ >> Yeah huh > > Yeah indeed. > > ~v~~ I meant, "does, too". 01oo
From: Virgil on 31 Mar 2007 19:18
In article <460ee112(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > On Mar 31, 5:30 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> Virgil wrote: > > > >>> In standard mathematics, an infinite sequence is o more than a function > >>> whose domain is the set of naturals, no two of which are more that > >>> finitely different. The codmain of such a function need not have any > >>> particular structure at all. > >> That's a countably infinite sequence. Standard mathematics doesn't allow > >> for uncountable sequences like the adics or T-riffics, because it's been > >> politically agreed upon that we skirt that issue and leave it to the > >> clerics. > > > > That's false; > > Please elucidate on the untruth of the statement. It should be easy to > disprove an untrue statement. TO claims politics is involved, but offers no proof, so that rejection of his claim as unproven is justified. TO claims religion is involved, but offers no proof, so that rejection of his claim as unproven is justified. > > > people have examined all sorts of orderings, partial, > > total, and other. The fact that you prefer to remain ignorant of this > > does not mean the issue has been skirted by anyone other than > > yourself. > > > > There have always been religious and political pressures on this area of > exploration. TO claims politics AND religion is involved, but offers no proof, so that rejection of his claim as unproven is justified. > > > > Yes, I left out some details. All of them, in fact. |