The Definition of Points
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From: Tony Orlow on 30 Mar 2007 13:10 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. 01oo
From: Tony Orlow on 30 Mar 2007 13:11 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. 01oo
From: Tony Orlow on 30 Mar 2007 13:13 Lester Zick wrote: > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>> It's the same as Peano. >>> Not it isn't, Tony. Cumulative addition doesn't produce straight lines >>> or even colinear straight line segments. Some forty odd years ago at >>> the Academy one of my engineering professors pointed out that just >>> because there is a stasis across a boundary doesn't necessarily mean >>> that there is no flow across the boundary only that the net flow back >>> and forth is zero.I've always been impressed by the line of reasoning. >> The question is whether adding an infinite number of finite segments >> yields an infinite distance. > > I have no idea what you mean by "infinite" Tony. An unlimited number > of line segments added together could just as easily produce a limited > distance. > > ~v~~ Not unless the vast majority are infinitesimal. If there is a nonzero lower bound on the interval lengths, an unlimited number concatenated produces unlimited distance. 01oo
From: Tony Orlow on 30 Mar 2007 13:22 Lester Zick wrote: > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> Finite addition never produces infinites in magnitude any more than >>> bisection produces infinitesimals in magnitude. It's the process which >>> is infinite or infinitesimal and not the magnitude of results. Results >>> of infinite addition or infinite bisection are always finite. >>> >>>> Wrong. >>> Sure I'm wrong, Tony. Because you say so? >>> >> Because the results you toe up to only hold in the finite case. > > So what's the non finite case? And don't tell me that the non finite > case is infinite because that's redundant and just tells us you claim > there is a non finite case, Tony, and not what it is. > If you define the infinite as any number greater than any finite number, and you derive an inductive result that, say, f(x)=g(x) for all x greater than some finite k, well, any infinite x is greater than k, and so the proof should hold in that infinite case. Where the proof is that f(x)>g(x), there needs to be further stipulation that lim(x->oo: f(x)-g(x))>0, otherwise the proof is only valid for the finite case. That's my rules for infinite-case inductive proof. It's post-Cantorian, the foundation for IFR and N=S^L. :) >> You can >> start with 0, or anything in the "finite" arena, the countable >> neighborhood around 0, and if you add some infinite value a finite >> number of times, or a finite value some infinite number of times, you're >> going to get an infinite product. If your set is one of cumulative sets >> of increments, like the naturals, then any infinite set is going to >> count its way up to infinite values. > > Sure. If you have infinites to begin with you'll have infinites to > talk about without having to talk about how the infinites you > have to talk about got to be that way in the first place. > > ~v~~ Well sure, that's science. Declare a unit, then measure with it and figure out the rules or measurement, right? 01oo
From: MoeBlee on 30 Mar 2007 13:23
On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote: > They > introduce the von Neumann ordinals defined solely by set inclusion, By membership, not inclusion. > and > yet, surreptitiously introduce the notion of order by means of this set. "Surreptitiously". You don't know an effing thing you're talking about. Look at a set theory textbook (such as Suppes's 'Axiomatic Set Theory') to see the explicit definitions. MoeBlee |