The Definition of Points
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From: Tony Orlow on 31 Mar 2007 20:55 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]. 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. What is a "number"? Good question. It's really the symbolic representation of a quantity. That's why folk like Han and WM discount unrepresentable numbers. I don't. I allow infinite strings, like the T-riffics and adics, and the uncountable sequence of the real H-riffics. tony.
From: Tony Orlow on 31 Mar 2007 20:56 Lester Zick wrote: > 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~~ It tells us "actual" means "uncountable" in the context of "infinite". 01oo
From: Tony Orlow on 31 Mar 2007 20:57 Lester Zick wrote: > 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~~ Brian feels better. That's what really matters, to me at least... 01oo
From: Tony Orlow on 31 Mar 2007 20:59 Lester Zick wrote: > On Fri, 30 Mar 2007 12:27:40 -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: >>> >>>>> Just ask yourself, Tony, at what magic point do intervals become >>>>> infinitesimal instead of finite? Your answer should be magnitudes >>>>> become infintesimal when subdivision becomes infinite. >>>> Yes. >>> Yes but that doesn't happen until intervals actually become zero. >>> >>>> But the term >>>>> "infinite" just means undefined and in point of fact doesn't become >>>>> infinite until intervals become zero in magnitude. But that never >>>>> happens. >>>> But, but, but. No, "infinite" means "greater than any finite number" and >>>> infinitesimal means "less than any finite number", where "less" means >>>> "closer to 0" and "more" means "farther from 0". >>> Problem is you can't say when that is in terms of infinite bisection. >>> >>> ~v~~ >> Cantorians try with their lame "aleph_0". Better you get used to the >> fact that there is no more a smallest infinity than a smallest finite, >> largest finite, or smallest or largest infinitesimal. Those things >> simply don't exist, except as phantoms. > > Does anyone really care? > > ~v~~ You can only answer that for yourself. What was the topic again? 01oo
From: Virgil on 31 Mar 2007 20:59
In article <460ee90d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460e56a5(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > > > >>> But all other mathematical objects are equally fantastic, having no > >>> physical reality, but existing only in the imagination. So any statement > >>> of mathematical existence is always relative to something like a system > >>> of axioms. > >> Sure, but the question is whether any such assumption of existence > >> introduces nonsense into your system. > > > > It has in each of TO's suggested systems so far. > > > > If thou so sayest, Sire. > > >> With the very basic assumption > >> that subtracting a positive amount from anything makes it less > > > > That presumes at least a definition of "positive" and a definition of > > "amount" and a definition of "subtraction" and a definition of "less" > > before it makes any sense at all. > > Yes, it does. Too late. Such definition have to precede, not follow, the claims. |