The Definition of Points
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From: Lester Zick on 1 Apr 2007 19:17 On Sat, 31 Mar 2007 18:36:28 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 30 Mar 2007 12:10:12 -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: >>>> >>>>>>> Their size is finite for any finite number of subdivisions. >> >>>>>> And it continues to be finite for any infinite number of subdivisions >>>>>> as well.The finitude of subdivisions isn't related to their number but >>>>>> to the mechanical nature of bisective subdivision. >>>>>> >>>>> Only to a Zenoite. Once you have unmeasurable subintervals, you have >>>>> bisected a finite segment an unmeasurable number of times. >>>> Unmeasurable subintervals? Unmeasured subintervals perhaps. But not >>>> unmeasurable subintervals. >>>> >>>> ~v~~ >>> Unmeasurable in the sense that they are nonzero but less than finite. >> >> Then you'll have to explain how the trick is done unless what you're >> really trying to say is dr instead of points resulting from bisection. >> I still don't see any explanation for something "nonzero but less than >> finite". What is it you imagine lies between bisection and zero and >> how is it supposed to happen? So far you've only said 1/00 but that's >> just another way of making the same assertion in circular terms since >> you don't explain what 00 is except through reference to 00*0=1. >> >> ~v~~ > >But, I do. > >I provide proof that there exists a count, a number, which is greater >than any finite "countable" number, for between any x and y, such that >x<y, exists a z such that x<z and z<y. No finite number of intermediate >points exhausts the points within [x,z], no finite number of >subdivisions. So, in that interval lie a number of points greater than >any finite number. Call |R in (0,1]| "Big'Un" or oo., Well you'll just have to excuse me, Tony, if I don't quite seem to get it unless in accordance with me you take 00 to represent a number of infinitesimals in which case your so called "points" are not points at all but infinitesimals intead. > and move on to the >next conclusion....each occupies how m,uch of that interval? dr. ~v~~
From: Lester Zick on 1 Apr 2007 19:18 On 31 Mar 2007 16:56:16 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >On 1 Apr, 00:36, Tony Orlow <t...(a)lightlink.com> wrote: >> Lester Zick wrote: >> > On Fri, 30 Mar 2007 12:10:12 -0500, Tony Orlow <t...(a)lightlink.com> >> > wrote: >> >> >> Lester Zick wrote: >> >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <t...(a)lightlink.com> >> >>> wrote: >> >> >>>>>> Their size is finite for any finite number of subdivisions. >> >> >>>>> And it continues to be finite for any infinite number of subdivisions >> >>>>> as well.The finitude of subdivisions isn't related to their number but >> >>>>> to the mechanical nature of bisective subdivision. >> >> >>>> Only to a Zenoite. Once you have unmeasurable subintervals, you have >> >>>> bisected a finite segment an unmeasurable number of times. >> >>> Unmeasurable subintervals? Unmeasured subintervals perhaps. But not >> >>> unmeasurable subintervals. >> >> >>> ~v~~ >> >> Unmeasurable in the sense that they are nonzero but less than finite. >> >> > Then you'll have to explain how the trick is done unless what you're >> > really trying to say is dr instead of points resulting from bisection. >> > I still don't see any explanation for something "nonzero but less than >> > finite". What is it you imagine lies between bisection and zero and >> > how is it supposed to happen? So far you've only said 1/00 but that's >> > just another way of making the same assertion in circular terms since >> > you don't explain what 00 is except through reference to 00*0=1. >> >> > ~v~~ >> >> But, I do. >> >> I provide proof that there exists a count, a number, which is greater >> than any finite "countable" number, for between any x and y, such that >> x<y, exists a z such that x<z and z<y. No finite number of intermediate >> points exhausts the points within [x,z], no finite number of >> subdivisions. So, in that interval lie a number of points greater than >> any finite number. Call |R in (0,1]| "Big'Un" or oo., and move on to the >> next conclusion....each occupies how m,uch of that interval? >> >> 01oo > >So.. you (correctly) note that there are not a finite "number" of >reals in [0,1]. You think this "proves" that there exists an infinite >"number". Why? (And, what is your definition of "number")? And what is your definition of "infinite"? ~v~~
From: Lester Zick on 1 Apr 2007 19:21 On Sat, 31 Mar 2007 22:42:19 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <460f1ef1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >> > In article <460f0317(a)news2.lightlink.com>, >> > Tony Orlow <tony(a)lightlink.com> wrote: >> > >> > >> >> There are not zero, nor any finite number of reals in (0,1]. >> > >> > There are every finite and more of reals in (0,1]. >> >> You mean any sequential ordering of the reals in (0,1] will contain >> elements in finite positions, plus more. Same thang. Tru dat, yo. > >What I mean, and what TO misrepresents me to mean have nothing in common. Yeah well I'd like to say that sentence reads just like you have more in common with great apes than anyone else except I have no interest in further demeaning great apes. ~v~~
From: Lester Zick on 1 Apr 2007 19:23 On 1 Apr 2007 03:49:59 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >On 1 Apr, 01:55, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >> > On 1 Apr, 00:36, Tony Orlow <t...(a)lightlink.com> wrote: >> >> Lester Zick wrote: >> >>> On Fri, 30 Mar 2007 12:10:12 -0500, Tony Orlow <t...(a)lightlink.com> >> >>> wrote: >> >>>> Lester Zick wrote: >> >>>>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <t...(a)lightlink.com> >> >>>>> wrote: >> >>>>>>>> Their size is finite for any finite number of subdivisions. >> >>>>>>> And it continues to be finite for any infinite number of subdivisions >> >>>>>>> as well.The finitude of subdivisions isn't related to their number but >> >>>>>>> to the mechanical nature of bisective subdivision. >> >>>>>> Only to a Zenoite. Once you have unmeasurable subintervals, you have >> >>>>>> bisected a finite segment an unmeasurable number of times. >> >>>>> Unmeasurable subintervals? Unmeasured subintervals perhaps. But not >> >>>>> unmeasurable subintervals. >> >>>>> ~v~~ >> >>>> Unmeasurable in the sense that they are nonzero but less than finite. >> >>> Then you'll have to explain how the trick is done unless what you're >> >>> really trying to say is dr instead of points resulting from bisection. >> >>> I still don't see any explanation for something "nonzero but less than >> >>> finite". What is it you imagine lies between bisection and zero and >> >>> how is it supposed to happen? So far you've only said 1/00 but that's >> >>> just another way of making the same assertion in circular terms since >> >>> you don't explain what 00 is except through reference to 00*0=1. >> >>> ~v~~ >> >> But, I do. >> >> >> I provide proof that there exists a count, a number, which is greater >> >> than any finite "countable" number, for between any x and y, such that >> >> x<y, exists a z such that x<z and z<y. No finite number of intermediate >> >> points exhausts the points within [x,z], no finite number of >> >> subdivisions. So, in that interval lie a number of points greater than >> >> any finite number. Call |R in (0,1]| "Big'Un" or oo., and move on to the >> >> next conclusion....each occupies how m,uch of that interval? >> >> >> 01oo >> >> > So.. you (correctly) note that there are not a finite "number" of >> > reals in [0,1]. You think this "proves" that there exists an infinite >> > "number". Why? (And, what is your definition of "number")? >> >> > -- >> > mike. >> >> There are not zero, nor any finite number of reals in (0,1]. > >OK so far. > >> There are >> more reals than either of those, an infinite number, farther from 0 than >> can be counted. If there were a finite number, then some finite number >> of intermediate points would suffice, but that leaves intermediate >> points unincluded. > >OK, so there are *not* a finite number of reals. > >Now, apparently, this *proves* that this thing called "BigUn" exists >to denote how many reals there are. And, apparently, this BigUn >behaves just like those good old finite numbers we're used to. We can >perform all the usual arithmetical operations on it. However, it's not >defined in any way other than "it's larger than finite". All we know >about it is that it's a "symbolic representation of quantity" and that >it's "larger than any finite". And yet we can do arithmetic with it >like it was a natural number. And yet modern mathematikers like Bob make straight lines out of points without blushing. Go figure. >Do you not see *any* problem with this picture? ~v~~
From: Lester Zick on 1 Apr 2007 19:24
On Sat, 31 Mar 2007 18:40:56 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 30 Mar 2007 12:11:23 -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: >>>> >>>>>> Equal subdivisions. That's what gets us cardinal numbers. >>>>>> >>>>> Sure, n iterations of subdivision yield 2^n equal and generally mutually >>>>> exclusive subintervals. >>>> I don't know what you mean by mutually exclusive subintervals. They're >>>> equal in size. Only their position differs in relation to one another. >>>> >>>> ~v~~ >>> Mutually exclusive intervals : intervals which do not share any points. >> >> What points? We don't have any points not defined through bisection >> and those intervals do share the endpoints with consecutive segments. >> >> ~v~~ > >Okay, lay off the coffee. > >Sure. Now subdivide the line so that the left endpoint is always >included and the right never. [x,y). Then each is mutually exclusive of >all others. Except the ones in the middle. ~v~~ |