The Definition of Points
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From: Tony Orlow on 22 Mar 2007 21:15 Lester Zick wrote: > On Thu, 22 Mar 2007 17:15:35 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Wed, 21 Mar 2007 22:24:22 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Wed, 21 Mar 2007 07:40:46 -0400, Bob Kolker <nowhere(a)nowhere.com> >>>>> wrote: >>>>> >>>>>> Tony Orlow wrote: >>>>>>> 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. :) >>>>>> Points (taken individually or in countable bunches) have measure zero. >>>>> They probably also have zero measure in uncountable bunches, Bob. At >>>>> least I never heard that division by zero was defined mathematically >>>>> even in modern math per say. >>>>> >>>>> ~v~~ >>>> Purrrrr....say! Division by zero is not undefinable. One just has to >>>> define zero as a unit, eh? >>> A unit of what, Tony? >>> >>>> Uncountable bunches certainly can attain nonzero measure. :) >>> Uncountable bunches of zeroes are still zero, Tony. >>> >>> ~v~~ >> Infinitesimal units can be added such that an infinite number of them >> attain finite sums. > > And since when exactly, Tony, do infinitesimals equal zero pray tell? > > ~v~~ Only in the "standard" universe, Lester. 01oo
From: Tony Orlow on 22 Mar 2007 21:28 Mike Kelly wrote: > 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. > Nice out. It has to do with the relationship between counting and the continuum. Do you invest much time in generalization? You might find it rewarding... >> 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. > Yeah, I know, see? .... >> That's not cardinality, of course, > > Of course. You're chronically incapable of sticking to the subject at > hand. > What was the subject again? Oh yeah, the difference between math based on points vs. lines. And, your point was....? >> so it's not an answer to CH within ZFC. > > It's not an answer to CH anywhere. > It's a system that makes the question obsolete, but whatever... >> 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? > See above. >> 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. > It probably doesn't. Robinson was much more tactful than I. > And, again, none of this has *anything whatsoever* to do with > geometry. > Uh, sure, if you say so. >>> 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? > In the sense of giving relative measure to sets beyond those which occupy some finite measure in a given finite interval. Let's not get too deep into that now. This was supposed to be about p[recedence of points vs. lines. >>>>> 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. > Riete! antonio.
From: Virgil on 22 Mar 2007 23:29 In article <46032753(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. TO, as usual, has cart-before-horse-itis. Sets can be a foundation of geometry but not vice versa. CH is purely about 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. The IFR is another of TO's delusions. > > >>> 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. Since TO steadfastly rejects the implications of his own assumptions, he guarantees that he will never reach that goal.
From: Mike Kelly on 23 Mar 2007 04:10 On 23 Mar, 01:28, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > 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. > > Nice out. It has to do with the relationship between counting and the > continuum. Do you invest much time in generalization? You might find it > rewarding... Nothing you say has anything to do with GCH, either. > >> 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. > > Yeah, I know, see? .... > > >> That's not cardinality, of course, > > > Of course. You're chronically incapable of sticking to the subject at > > hand. > > What was the subject again? Oh yeah, the difference between math based > on points vs. lines. And, your point was....? MK: You can't state a limitation of current axiomatisations of gemoetry. TO: They don't consider that there are oo points in a unit interval. MK: "Declaring" that there are oo points in a unit interval leads to no new geometrical theorems. TO: Consider CH. Declaring that there are oo points in a unit interval resolves/answers/circumvents CH. MK: It has nothing to do with CH. Moreover, CH has nothing to do with geometry. TO: Yes it does. MK: No, it doesn't. TO: I know it doesn't. What is your point? My point : Nothing you are saying is an "answer" to CH and *even if it were* it wouldn't be *anything whatsoever* to do with geometry. As such you haven't demonstrated a "limitation" with current axiomatisations of geometry. > >> 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. > > It probably doesn't. Robinson was much more tactful than I. Not the first difference between you two that springs to mind, I must admit. Do you even understand what I was saying here? Having a system with a "full spectrum of infinities" doesn't automatically obselete CH. Obviously, you don't have anything even close to a coherent set of ideas to make up your "system" and are nowhere near capable of beginning the proces of formalising it. So, I find it a little strange that so cavalierly dismiss CH, asserting that in your "system" CH will be "obselete", when it isn't obselete in NSA, a hugely more developed theory with "a full spectrum of infinities". > > And, again, none of this has *anything whatsoever* to do with > > geometry. > > Uh, sure, if you say so. Yes, I do say so. The *entire* point of this subthread was that your "answer" to CH is nothing to do with geometry so I'm glad you concede the point, finally. > >>> 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? > > In the sense of giving relative measure to sets beyond those which > occupy some finite measure in a given finite interval. Let's not get too > deep into that now. This was supposed to be about p[recedence of points > vs. lines. This was supposed to be about you stating what the "limitations" of the current axiomatisation of geometry are. You haven't done so. -- mike.
From: Bob Cain on 23 Mar 2007 05:16
Lester Zick wrote: > You know I'm beginning to get the distinctly unpleasant impression > your putatuve psychological expose of me is more self portrait than > anything else. The rest of us, that consensus you believe it your mission to tweak, consider PD's sketch a very witty and perfectly appropriate portrait of you and your vanity threads, Lester. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein |