The Definition of Points
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From: Lester Zick on 22 Mar 2007 19:11 On Thu, 22 Mar 2007 17:31:19 +0000 (UTC), stephen(a)nomail.com wrote: >In sci.math Mike Kelly <mikekellyuk(a)googlemail.com> wrote: >> On 22 Mar, 16:38, Tony Orlow <t...(a)lightlink.com> wrote: >>> Mike Kelly wrote: >>> >>> > Firstly, the continuum hypothesis is nothing to do with geometry. >>> >>> It does, if sets are combined with measure and a geometrical >>> representation of the question considered. Is half an infinity less than >>> itself? Geometrically, yes, the reals in (0,1] are half the reals in (0,2]. > >> Everything you just said has nothing to do with the continuum >> hypothesis. You're terminally confused. > >> As near as I can tell, your mish-mash of ideas all basically boil down >> to asserting "the 'number' of reals/points in an interval/line segment >> = the Lebesgue measure". What does asserting that do for us? Not much. > >>> > Secondly, division is not defined for infinite cardinal numbers. >>> >>> I'm not interested in cardinality, but a richer system of infinities, >>> thanks. > >> You brought up the continuum hypothesis. Next post, you say you don't >> want to talk about cardinality. Risible. > >Tony thinks that the continuum hypothesis is independent >of cardinality, and that he can just plug in his own notions of "size". >That is about as stupid as thinking that > sqrt(3)^4+sqrt(4)^4=sqrt(5)^4 >is a refutation of Fermat's Last Theorem. Fermat's Last Theorem is about as trivial as Stephen's observations. ~v~~
From: Lester Zick on 22 Mar 2007 19:32 On 22 Mar 2007 11:13:08 -0700, "Brian Chandler" <imaginatorium(a)despammed.com> wrote: >Lester Zick wrote: >> On 22 Mar 2007 06:24:46 -0700, "PD" <TheDraperFamily(a)gmail.com> wrote: >> >On Mar 21, 6:00 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > ><snip-snop> > >> As previously noted brevity is the soul of wit and you're neither. > >Neither what? I mean, if neither this nor that, what are this and >that? Neither witty nor brief. I'm appalled a passed zen master like you couldn't decipher a trivial bon mot. Perhaps you'd care to try the first problem in "Takin Out the Trash"? I mean just for practice as I expect you could decipher it in a trice.Just another example of my wit, Brian. I'm sure Draper would prefer it if I were witless so he could keep up but there it is. >(Don't bother to answer if it takes too much time out of your >lecturing schedule...) Doesn't so much take time out of my lecturing schedule, Brian, it is my lecturing schedule. Alas one must do what one can to convert the heathen. Last time I asked DvdM what a clock was he got all huffy and departed for parts uncharted. ~v~~
From: Mike Kelly on 22 Mar 2007 19:38 On 22 Mar, 21:42, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 22 Mar, 16:38, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> On 22 Mar, 13:03, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > >>>>> On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> Mike Kelly wrote: > >>>>>>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>> step...(a)nomail.com wrote: > >>>>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>>>> step...(a)nomail.com wrote: > >>>>>>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>>>>>> step...(a)nomail.com wrote: > >>>>>>>>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>>>>>>>> PD wrote: > >>>>>>>>>>>>>>> No one says a set of points IS in fact the constitution of physical > >>>>>>>>>>>>>>> object. > >>>>>>>>>>>>>>> Whether it is rightly the constitution of a mentally formed object > >>>>>>>>>>>>>>> (such as a geometric object), that seems to be an issue of arbitration > >>>>>>>>>>>>>>> and convention, not of truth. Is the concept of "blue" a correct one? > >>>>>>>>>>>>>>> PD > >>>>>>>>>>>>>> The truth of the "convention" of considering higher geometric objects to > >>>>>>>>>>>>>> be "sets" of points is ascertained by the conclusions one can draw from > >>>>>>>>>>>>>> that consideration, which are rather limited. > >>>>>>>>>>>>> How is it limited Tony? Consider points in a plane, where each > >>>>>>>>>>>>> point is identified by a pair of real numbers. The set of > >>>>>>>>>>>>> points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. > >>>>>>>>>>>> That's a very nice circle, Stephen, very nice.... > >>>>>>>>>>> Yes, it is a circle. You knew exactly what this supposedly > >>>>>>>>>>> "limited" description was supposed to be. > >>>>>>>>>>>>> In what way is this description "limited"? Can you provide a > >>>>>>>>>>>>> better description, and explain how it overcomes those limitations? > >>>>>>>>>>>> There is no correlation between length and number of points, because > >>>>>>>>>>>> there is no workable infinite or infinitesimal units. Allow oo points > >>>>>>>>>>>> per unit length, oo^2 per square unit area, etc, in line with the > >>>>>>>>>>>> calculus. Nuthin' big. Jes' give points a size. :) > >>>>>>>>>>> How is that a limitation? You knew exactly what shape the > >>>>>>>>>>> set of points described. There is no feature of the circle > >>>>>>>>>>> that cannot be determined by the above description. There is > >>>>>>>>>>> no need to correlate length and number of points. Neither > >>>>>>>>>>> Euclid or Hilbert ever did that. > >>>>>>>>>> Gee, I guess it's a novel idea, then. That might make it good, and not > >>>>>>>>>> necessarily bad. Hilbert also didn't bother to generalize his axioms to > >>>>>>>>>> include anything but Euclidean space, which is lazy. He's got too many > >>>>>>>>>> axioms, and they do too little. :) > >>>>>>>>> Then you should not be complaining about the "set of points" approach > >>>>>>>>> to geometry and instead should be complaining about all prior approaches > >>>>>>>>> to geometry. Apparently they are all "limited" to you. Of course > >>>>>>>>> you cannot identify any actual limitation, but that is par for the course. > >>>>>>>> I wouldn't say geometry is perfected yet. > >>>>>>> And yet you remain incapable of stating what these "limitations" are. > >>>>>>> Why is that? Could it be because you can't actually think of any? > >>>>>>> -- > >>>>>>> mike. > >>>>>> I already stated that the divorce between infinite set size and measure > >>>>>> of infinite sets of points is a limitation, and indicated a remedy, but > >>>>>> I don't expect you to grok that this time any better than in the past. > >>>>>> Keep on strugglin'.... > >>>>>> tony. > >>>>> What theorems can't be proved with current axiomatisations of geometry > >>>>> but can be with the addition of axiom "there are oo points in a unit > >>>>> interval"? Anything more interesting than "there are 2*oo points in an > >>>>> interval of length 2"? > >>>>> -- > >>>>> mike. > >>>> Think "Continuum Hypothesis". If aleph_1 is the size of the set of > >>>> finite reals, is aleph_1/aleph_0 the size of the set of reals in the > >>>> unit interval? Is that between aleph_0 and aleph_1? Uh huh. > >>> Firstly, the continuum hypothesis is nothing to do with geometry. > >> It does, if sets are combined with measure and a geometrical > >> representation of the question considered. Is half an infinity less than > >> itself? Geometrically, yes, the reals in (0,1] are half the reals in (0,2]. > > > Everything you just said has nothing to do with the continuum > > hypothesis. You're terminally confused. > > So, CH doesn't have to do with whether c, the number of reals, is > greater than aleph_1, the next transfinite cardinal after aleph_0? If > there is a cardinal number less than the [cardinal] number of all reals, and > greater than the [cardinal] number of all naturals, then c>aleph_1, right? CH : c = aleph_1, yes. Nothing you said had anything to do with cardinality so it had nothing to do with CH. > If the > number of reals overall is the number of reals per unit interval times > the number of unit intervals (times aleph_0, that is), then the number > of reals per unit interval qualifies as a value between the number of > naturals and the number of reals. You're not talking about cardinality here so, like I said, you are NOT talking about anything to do with the continuum hypothesis. > That's not cardinality, of course, Of course. You're chronically incapable of sticking to the subject at hand. > so it's not an answer to CH within ZFC. It's not an answer to CH anywhere. > The system I'm talking about doesn't require a hypothesis about this. So why are you pretending you're talking about the continuum hypothesis? > The answer is clear. There are > a full spectrum of infinities above and below c. This doesn't answer whether there is a cardinality between that of the naturals and that of the reals. Do you think the "answer" to CH is clear in Robinson's NSA, for example? In fact, it isn't. If CH is assumed then any choice of ultrafilter leads to an (isomorphically) identical quotient field. If ~CH is assumed then there are choices of ultrafilter that lead to different fields which are not isomorphic. If just using ZFC, one cannot prove one way or the other. I don't know if you consider NSA to have a "full spectrum of infinities" but it doesn't provide an "answer" to CH. And, again, none of this has *anything whatsoever* to do with geometry. > > As near as I can tell, your mish-mash of ideas all basically boil down > > to asserting "the 'number' of reals/points in an interval/line segment > > = the Lebesgue measure". What does asserting that do for us? Not much. > > It goes a bit beyond Lebesgue measure, but I don't expect you to see that. Your "system" is nowhere near as well defined as Lebesgue measure. In what way does it go beyond it? > >>> Secondly, division is not defined for infinite cardinal numbers. > >> I'm not interested in cardinality, but a richer system of infinities, > >> thanks. > > > You brought up the continuum hypothesis. Next post, you say you don't > > want to talk about cardinality. Risible. > > CH is a question in ZFC. The answer lies outside ZFC. Risible. -- mike.
From: Tony Orlow on 22 Mar 2007 21:03 Virgil wrote: > In article <4602b106(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> Firstly, the continuum hypothesis is nothing to do with geometry. >> It does, if sets are combined with measure and a geometrical >> representation of the question considered. > > The CH still has nothing directly to do with geometry. > True, But, where geometry has to do with sets of atomic points and measure, well, it has something to say about the infinity of sets. > > >> Is half an infinity less than >> itself? Geometrically, yes, the reals in (0,1] are half the reals in (0,2]. > > Not in any sense recognized by the CH. Not by bijection alone. By IFR it's a fact. >>> Secondly, division is not defined for infinite cardinal numbers. >> I'm not interested in cardinality, but a richer system of infinities, >> thanks. > > Which supposedly richer system is still so poor that it it does not > exist. Other than as one of TO's pipe dreams. Not yet as a complete replacement for ZFC, but that wasn't built i na day, or a few years, either. :)
From: Tony Orlow on 22 Mar 2007 21:12
Lester Zick wrote: > On Thu, 22 Mar 2007 17:14:38 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Wed, 21 Mar 2007 22:45:54 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Wed, 21 Mar 2007 14:17:16 -0500, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>> >>>>>> It states the specific infinite number of points in the unit interval, >>>>>> say, on the real line. >>>>> And what real line would that be, Tony? >>>>> >>>>> ~v~~ >>>> The one that fully describes the real numbers. >>> You mean a straight line that describes curves exactly? Or some curve >>> that describes straight lines exactly? >>> >> There is no straight line, but only the infinitesimally curved. :D > > What evidence do you have to support your opinion, Tony? Straight > lines have zero curvature. > What evidence would you like for the difference between zero and an infinitesimal? Not something finite, I hope.... >>>> Like, duh! The one that >>>> exists. >>> Except there is no such line, Tony. At least none that describes both >>> curves and straight lines together exactly.And if you don't believe me >>> Bob Kolker has acknowledged the point previously. >>> >> Well, if Bob says so, then.... > > I might agree if Bob hadn't gone out of his way to say so. It was > uncharacteristic of him to say so. But the very fact that he could see > so and said so inclines me to his opinion rather than yours. > So be it. >>>> E R >>>> 0eR >>>> 1eR >>>> 0<1 >>>> xeR ^ yeR ^ x<y -> EzeR x<z ^ z<y >>> Very fanciful, Tony. You mean if you know the approximation for pi >>> lies between 3 and 4 on a straight line pi itself does too? >> For instance. If you can get arbitrarily close to pi without leaving the >> line, then it resides on the line. Otherwise it would be some distance >>from the line, and there would be a lower limit to your approximation. > > What makes you think pi resides on straight lines instead of circles? It is between 3 and 4. > Or do you think straight lines reside on curves? They may intersect... Or do you think pi > resides on both and a circle is equal to straight line approximations? An infinitely regressive one, perchance. > Otherwise it would indeed be at some distance from the straight line > and there would be no point on the straight line corresponding to pi. > Then one could not point out arbitrarily close points to the desired irrational or transcendental, which all reside on the line. The distance from that point to the line would be some lower limit. >>> You see, Tony, this is the basic reason I refuse to be drawn into >>> discussion on collateral mathematical issues as interesting as they >>> might be. I can't even get the most elementary point across even to >>> those supposedly paying attention to what I say. >>> >>> ~v~~ >> Try me. > > Already have. Don't know what else to say. > > ~v~~ You'll think of something. 01oo |