An uncountable countable set
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From: Lester Zick on 4 Oct 2006 13:10 On 3 Oct 2006 23:15:04 -0700, imaginatorium(a)despammed.com wrote: > >David R Tribble wrote: >> Tony Orlow wrote: >> >> On the other hand >> >> I don't know why I said "neither can the reals". In any case, the only >> >> way the ordinals manage to be "well ordered" is because they're defined >> >> with predecessor discontinuities at the limit ordinals, including 0. >> >> That doesn't seem "real" >> > >> >> Virgil wrote: >> >> In what sense of "real". There are subsets of the reals which are order >> >> isomorphic to every countable ordinal, including those with limit >> >> ordinals, so until one posits uncountable ordinals there are no problems. >> > >> >> Tony Orlow wrote: >> > In the sense that the real world is continuous, and you don't just have >> > these beginnings with nothing before them. The real line is a line, with >> > each point touching two others. >> >> That's a neat trick, considering that between any two points there is >> always another point. An infinite number of points between any two, >> in fact. So how do you choose two points in the real number line >> that "touch"? > >Me! Me!! I can do this one!! > >Er, well, here's how. You choose a first point (P). Easy, yes? Well perhaps not quite so easy as you might imagine, Brian. Perhaps you'd care to spell out exactly how you choose a first point that's so easy? Now please don't try to beg off from the problem under the pretext that actually choosing points is easy since it's an excercise in physics where you don't excel but not in the zen of mathematics where you don't seem to excel either. Just tell us how the trick is done that's so easy that's anything other than you're saying it's so. [. . .] > then we would have chosen it before lunchtime, >but now it's time for lunch. Contradiction. > >We have shown two adjacent points Pz and Q. You can't argue about how >far apart they are (obviously an infinitesimal, since infinitesimals >have the desired properties in all context), because I've gone to >lunch. So infinitesimals are infinitesimal since they're infinitesimals to begin with because quite possibly you are already out to lunch? ~v~~
From: Lester Zick on 4 Oct 2006 13:20 On Tue, 03 Oct 2006 21:21:27 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <45231438(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > > >> > The property of not being an infinite natural holds for the first >> > natural, and holds for the successor of each non-infinite natural, so >> > that it must hold for ALL naturals. >> >> It holds for all finite naturals > >It holds for ALL naturals, as that " inductiveness" is an essential part >of the definition of the naturals. > >Anything without that property, whatever it may be, is not the set of >naturals, though it may contain the naturals as a proper subset. One is forced to wonder though exactly how one gets from "inductive addition" to "inductiveness" in general. Certainly the peano axioms assume the former for definition of natural numbers but the latter is known to be incorrect in the case of post-hoc-propter-hoc fallacies. ~v~~
From: Randy Poe on 4 Oct 2006 13:21 Lester Zick wrote: > On 4 Oct 2006 05:19:38 -0700, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: > > >Tony Orlow wrote: > > [. . .] > > >Note : I agree with those who say it makes no sense in physical terms > >to have an infinite number of balls. But mathematics is an idealisation > >so it can make sense to talk about the infinite, even if it is > >physically impossible. > > So physics has to make sense but math doesn't? Math has to be logical. It doesn't have to be physically realizable. > I think you'd find > plenty of quantum theorists, hyperdimensionalists and relativists > who'd disagree. I think you underestimate their sophistication. Physicists learn at a young age that things can exist in mathematics that are not physically realizable and yet are very useful. Point masses and continuous mass distributions for instance. And they also realize that things can exist in mathematics that aren't even approximations of a physical realizable. That aren't physically sensible in other words. You are conflating "physically sensible" with "logical", the same way you mistake "agrees with my untrained common sense" with "sensible". - Randy
From: Lester Zick on 4 Oct 2006 13:43 On 3 Oct 2006 15:48:15 -0700, "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote: >Lester Zick wrote: >> On 3 Oct 2006 13:46:18 -0700, imaginatorium(a)despammed.com wrote: >> >> >Lester Zick wrote: >> >> On 3 Oct 2006 10:13:54 -0700, imaginatorium(a)despammed.com wrote: >> >> >> >> >Lester Zick wrote: >> >> > >> >> ><snip lesser maundering...> >> >> > >> >> >> But if the limit concept is the only sensible way to approach the >> >> >> infinite, one is left to ponder whether the limit concept is infinite >> >> >> and if not how a limits to the limit concept can be approached? >> >> > >> >> >Beautifully put, Lester, but surely the only way would be by >> >> >regression? >> >> >> >> Naturally by finite tautological regression to self contradictory >> >> alternatives. Or subdivision. >> > >> >Hmm. Not sure what self-contradictory alternatives are. >> >> "Contradiction of contradiction" for one is the self contradictory >> tautological alternative to "contradiction". "Different from >> differences" for another is the self contradictory tautological >> alternative to "differences". And "not not" for yet another is the >> self contradictory tautological alternative to "not". >> >> > How can >> >something be alternative to something that's self-contradictory? >> >> Tautologically take what is "not" what's generally self contradictory. >> However you have to be sure of what's self contradictory in general >> scientific terms because there are general and particular self >> contradictions. If one simply asserts "Susan is not Susan" the self >> contradiction is particular and not general. So just denying it >> results in regression to at best a particular truth and possibly >> another particular self contradiction and not a general truth since >> "Susan not Susan" and "not (Susan not Susan)" don't necessarily >> exhaust all possibilities for truth between them. >> >> > Oh, >> >but I see, we could just use subdivision. Wow - that really would >> >_never_ have occurred to me. But I'm sorry, I don't quite see what we >> >would be subdividing; is there any chance you could clarify? (Hope the >> >answer won't entail any of these self-contradictory alternatives...) >> >> Not if you employ subdivision to begin with, Brian. One can subdivide >> indefinitely and infinitesimally as through the bisection of an angle >> or straight line segment without violating containment and "infinity" >> is the number of those infinitesimals whereas one can never increment >> a number without either producing another finite number or violating >> containment. >> >> ~v~~ > >Hi, > >Tony, the "Factorial/Exponential Identity, Infinity", that was last >year on sic.math. In it, I showed an expression that was obviously >untrue in the finite, but appeared to be so in the infinite, for some >input values. > >I just read it, and let's see, here, hypermatrices, Stirling's numbers, >density of infinite bit strings, basically it is about density of >semi-infinite bit strings. I see a variety of avenues for research. > >We get to talking about the real numbers, which I call R. Sometimes >you want to talk about them as continuous, other times, basically >discrete. So, there are optional umlaut and bar accents, two dots for >one, and the bar for the other, and both for both, R bar umlaut, R bar, >R umlaut, blah blah blah, R dot dot, the bar is horizontal. > >Then, there are the natural numbers, for example N, the symbol used to >denote the set of numbers {0, 1, 2, 3, ...}. I call the natural >numbers N. > >N E N! > >That's says that the set N contains itself. It also says it is >contained by itself. Thus the direct product of infinitely many copies >is zero. > >That's why ZF is insufficient to be the foundation for some >mathematics, because it is so easy to comprehend. That's a pun about >comprehension. > >Mathematics works that way. > >I read Cornelius Lanczos' "Linear Differential Operators", let me tell >you, he doesn't much talk about linear differential operators. > >I wrote about the ball and vase, basically the faster they move the >harder it is to see them. > >Hi Lester, hey how's it going. As usual, we're sitting here on >sci.math. You want a countable uncountable set? Look at the >rationals. 2^|X|, they go. Hi, Ross. "Sitting" being the operative term I expect. "Thinking" perhaps being preferable. I don't want a countable uncountable set. In fact there isn't much in modern math that's both counter intuitive and not counter intuitive that I would want as I've never been a big fan of analyzing axiomatic intuition in counter intuitive terms. To be perfectly frank most of these guys appear pretty much stuck on stupid anyway. It's always appeared amazing that mathemagicians see nothing problematic is seriously discussing ideas they can only guess at and assume are true. No one can define infinity except in terms of counter intuitive this and counter intuitive that but every other word out of their mouths is infinite this and infinity that. Makes you kinda wonder what they think they're talking about. ~v~~
From: Lester Zick on 4 Oct 2006 13:44
On 3 Oct 2006 22:22:07 -0700, "David R Tribble" <david(a)tribble.com> wrote: >Tony Orlow wrote: >>> On the other hand >>> I don't know why I said "neither can the reals". In any case, the only >>> way the ordinals manage to be "well ordered" is because they're defined >>> with predecessor discontinuities at the limit ordinals, including 0. >>> That doesn't seem "real" >> > >Virgil wrote: >>> In what sense of "real". There are subsets of the reals which are order >>> isomorphic to every countable ordinal, including those with limit >>> ordinals, so until one posits uncountable ordinals there are no problems. >> > >Tony Orlow wrote: >> In the sense that the real world is continuous, and you don't just have >> these beginnings with nothing before them. The real line is a line, with >> each point touching two others. > >That's a neat trick, considering that between any two points there is >always another point. An infinite number of points between any two, >in fact. So how do you choose two points in the real number line >that "touch"? Probably the same way you choose a real number line to begin with: by guess and by golly. ~v~~ |