The Definition of Points
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From: Mike Kelly on 31 Mar 2007 19:56 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.
From: Lester Zick on 31 Mar 2007 19:56 On Fri, 30 Mar 2007 12:25:24 -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: >> >>>>> If n is >>>>> infinite, so is 2^n. If you actually perform an infinite number of >>>>> subdivisions, then you get actually infinitesimal subintervals. >>>> And if the process is infinitesimal subdivision every interval you get >>>> is infinitesimal per se because it's the result of a process of >>>> infinitesimal subdivision and not because its magnitude is >>>> infinitesimal as distinct from the process itself. >>> It's because it's the result of an actually infinite sequence of finite >>> subdivisions. >> >> And what pray tell is an "actually infinite sequence"? >> >>> One can also perform some infinite subdivision in some >>> finite step or so, but that's a little too hocus-pocus to prove. In the >>> meantime, we have at least potentially infinite sequences of >>> subdivisions, increments, hyperdimensionalities, or whatever... >> >> Sounds like you're guessing again, Tony. >> >> ~v~~ > >An actually infinite sequence is one where there exist two elements, one >of which is an infinite number of elements beyond the other. Which tells us what exactly, Tony, infinite sequences are infinite? ~v~~
From: Lester Zick on 31 Mar 2007 19:59 On 30 Mar 2007 21:17:38 -0700, "Brian Chandler" <imaginatorium(a)despammed.com> wrote: >> Under what definition of sequence? > >Oh come on... definition schmefinition. This is Tony's touchy-feely >statement of what he feels it would be for a sequence to be "actually >infinite". Actually. The same could be said for your touchy feely definitions, Brian. Six of one half dozen of the other. >You're just being disruptive, trying to inject some mathematics into >this stream of poetry... Mathematics? What mathematics did you have in mind exactly, Brian? SOAP operas? Zen? What pray tell? ~v~~
From: Tony Orlow on 31 Mar 2007 19:58 stephen(a)nomail.com wrote: > In sci.math Brian Chandler <imaginatorium(a)despammed.com> wrote: >> stephen(a)nomail.com wrote: >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>> If all other elements in the sequence are a finite number >>>> of steps from the start, and w occurs directly after those, then it is >>>> one step beyond some step which is finite, and so is at a finite step. >>> So you think there are only a finite number of elements between 1 and >>> w? What is that finite number? 100? 100000? 100000000000000000? >>> 98042934810235712394872394712349123749123471923479? Which one? > >> None of the ones you've mentioned. Although it is, of course, a >> perfectly ordinary natural number, in that one can add 1 to it, or >> divide it by 2, its value is Elusive. Only Tony could actually write >> it down. > > These Elusive numbers have amazing properties. According to > Tony, there are only a finite number of finite naturals. > There exists some finite natural Q such that the set > { 1,2,3,4,.... Q} > is the set of all finite natural numbers. But what of Q+1? > Well we have a couple of options: > a) Q+1 does not exist > b) Q+1 is not a finite natural number > c) { 1,2,3,4, ... Q} is not the set of all finite natural numbers > > Tony rejects all these options, and apparently has some fourth > Elusive option. > > Stephen > Oy. The "elusive" option is that there is no acceptable "size" for N. That was really hard to figure out after all this time... Tony
From: Tony Orlow on 31 Mar 2007 20:01
Bob Kolker wrote: > Tony Orlow wrote:>> >> >> Measure makes physics possible. > > On compact sets which must have infinite cardinality. > > The measure of a dense countable set is zero. > > Bob Kolker Yes, some finite multiple of an infinitesimal. Tony Orlow |