An uncountable countable set
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From: Tony Orlow on 9 Oct 2006 15:41 David R Tribble 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. > > Tony Orlow wrote: >> It holds for all finite naturals, > > ... because all naturals are finite ... > >> but if there are an infinite number of >> naturals generating using increment, > > ... because the increment operation is "applied" an infinite number > of times, "generating" an infinitude of finite naturals ... > >> then there are naturals which are >> the result of infinite increments, which must have infinite value. > > Where's your proof? > > What is an "infinite increment" (or "infinite successor")? > If there is a !number! n of successors, there exists a successor n steps ahead. If there are an infinite !number! n of successors, there is a successor n, an infinite number of steps ahead. If you increment a natural n times, you have added n to it. If successor is increment, and there are an infinite !number! of such increments, you have added this infinite number to your starting value. Adding an infinite number to a finite yields an infinite. Therefore, the infinite set includes infinite values.
From: Tony Orlow on 9 Oct 2006 15:56 Lester Zick wrote: > On Sun, 08 Oct 2006 22:07:23 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Sun, 08 Oct 2006 15:12:28 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Sat, 07 Oct 2006 23:45:23 -0400, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>> >>>>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>>>> Tony Orlow schrieb: >>>>>>> >>>>>>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>>>>>> Tony Orlow schrieb: >>>>>>>>> >>>>>>>>> >>>>>>>>>>>> Why not? Each and every number of the list terminates. That one is a number >>>>>>>>>>>> that does *not* terminate. >>>>>>>>>>>> >>>>>>>>>>>> > If you think that 0.111... is a number, but not in the list, >>>>>>>>>>> It is me who insists that it is not a representation of a number. >>>>>>>>>> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" >>>>>>>>>> number. It's still some kind of quantitative expression, even if it's >>>>>>>>>> unbounded. Would you agree that ...333>...111, given a digital number >>>>>>>>>> system where 3>1? >>>>>>>>> That is the similar to 0.333... > 0.111.... But all these >>>>>>>>> representations exist only potentially, in my opinion. The difference >>>>>>>>> is, that 0.333... can be shown to lie between two existing numbers, so >>>>>>>>> we can calculate with it, while for ...333 this cannot be shown. >>>>>>>> I think it can be shown to lie between ...111 and ...555, given that >>>>>>>> each digit is greater than the corresponding digit in the first, and >>>>>>>> less than the corresponding digit in the second. >>>>>>> Yes, but only if we define, for instance, >>>>>>> >>>>>>> A n eps |N : 111...1 < 333...3 where n digits are symbolized in both >>>>>>> cases. >>>>>>> >>>>>>> This approach would be comparable with the "measure" which gives >>>>>>> >>>>>>> A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. >>>>>>> >>>>>>> I don't know whether these definitions are of any use, but I am sure >>>>>>> that they are not less useful than Cantor's cardinality. >>>>>>> >>>>>>> Regards, WM >>>>>>> >>>>>>> . >>>>>>> >>>>>> My opinion about that is, if one wants to talk about what happens "at >>>>>> infinity", that's the way that makes sense, not the measureless way of >>>>>> abstract set theory. I trust limit concepts, but not limit ordinals. >>>>> Tony, would it be fair to characterize what you're trying to say as >>>>> that there is some kind of positive/negative crossover at infinity >>>>> such that {-00, . . .,-1, 0, +1, . . . +00}? I haven't really been >>>>> following this thread too closely so I'm trying to understand what >>>>> you're after here in basic terms instead of the exact arguments >>>>> involved. >>>>> >>>>> ~v~~ >>>> Hi Lester, how's thangs? >>> Hey, Tony, pretty much as usual. >>> >>>> I wasn't saying that right here, but agreeing with Wolfgang that limit >>>> concepts make sense, while the transfinitological approach doesn't. What >>>> I did say was that there are two ways to view the number line, one where >>>> oo and -oo are polar opposites, and a number circle where they are the >>>> same. >>> But there is no single real number line. There's a single rational/ >>> irrational line but not even a single transcendental line, Tony. So on >>> the surface I don't see what this speculation has to recommend it. And >>> more to the point I don't see any way to effect a crossover in >>> mechanical terms. >> I've certainly heard you discuss your views on the number line, and how >> pi lies on a curve and rationals lie on straight lines, etc. To me, it >> sounds like a matter of construction, or meaning of the number, but not >> one of raw quantity. In terms of raw quantity on the real number line, >> they all obey the law of trichotomy, for any a and b, either a=b, a>b, >> or a<b. So, it's a linear order. > > As far as transcendentals are concerned, Tony, the only thing that can > lie on a real number line in common with rationals/irrationals are > straight line segment approximations. That's the only linear order > possible. So either you give up transcendentals or a real number line. The trichotomy or real quantity itself defines a linear order. Each such value is greater than or less than every different value. Pi is transcendental - is it less than or greater than 3? Is there any doubt about that? > >>>> When it comes to reality, pretty much everything is in circles, >>>> including finite number systems so this model makes some sense, even if >>>> it doesn't in terms of limits of, say, powers. lim(x->-oo)=0 and >>>> lim(x->oo)=oo for n^x where n>1, so there the two are different. >>> Well in a way I'm not sure I disagree. I'm interested in this aspect >>> of the problem mainly because I think that perhaps the issue both of >>> you may be trying to address is the closing of otherwise open sets. >>> Remember x->0 or plus or minus 00 doesn't mean x ever gets there. >> Perhaps not, but if, say, y=1/x is both oo and -oo at x=0, and oo=-oo, >> then the function is continuous in every respect, which is what we might >> desire in such a fundamental algebraic relation. > > But for the division operation x never becomes zero. Which indicates > that there can be no plus or minus infinity and no continuity. Is 0 part of the continuum, or just another arbitrary "limit" discontinuity? When you ask yourself, "If I divide this finite space into individual points, how many will I have?", what answer do you get? How much of the space between 0 and 1 does each real in that interval occupy, and how many are there? > >>>> When it comes down to this argument, Wolfgang's argument, I agree with >>>> his logic concerning the naturals and the identity function between >>>> element count and value. >>> To me "element count" and the number of commas are the same. >> Sure, that sounds okay to me. In the naturals, the first is 1, the >> second 2, etc. What is the aleph_0th? > > I don't know what you're asking here, Tony. If there is no real number > line aleph there are no aleph ordinals either. There can be aleph > infinitesimals but that represents a continual process of subdivision > and not one of division in which case the ordinality would be one of > relation between various curvatures where straight lines would be > first or minimal and the ordinality of others judged i
From: Virgil on 9 Oct 2006 15:57 In article <452aa5c9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > If you increment a natural n times, you have added n to it. If successor > is increment, and there are an infinite !number! of such increments, you > have added this infinite number to your starting value. Adding an > infinite number to a finite yields an infinite. Therefore, the infinite > set includes infinite values. Without a limit ordinals in between or as the larger, one never can have two ordinals separated by infinitely many successors of the smaller. And as every natural is an ordinal, that scuttles TO's theories.
From: Tony Orlow on 9 Oct 2006 15:58 David R Tribble wrote: > Tony Orlow wrote: >>> That doesn't seem "real", and the axiom of choice aside, I don't see >>> there being any well ordering of the reals. The closest one can come is >>> the H-riffic numbers. :) > > David R Tribble wrote: >>> Hardly. The H-riffics are a simple countable subset of the reals. >>> Anyone mathematically inclined can come up with such a set. > > Tony Orlow wrote: >> You never paid enough attention to understand them. They cover the reals. > > They omit an uncountable number of reals. Any power of 3, for example, > which you never showed as being a member of them. Show us how 3 fits > into the set, then we'll talk about "covering the reals". > 3 is an unending string, just like 1/3 is in base-10. Rusin confirmed that about two years ago. But, you're right, I need to construct a formal proof of the equivalence between the H-riffics and the reals.
From: Tony Orlow on 9 Oct 2006 16:04
Virgil wrote: > In article <452a6847(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <4529afa4(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <45296779(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Virgil wrote: >>>>>>> In article <452946ad(a)news2.lightlink.com>, >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>>> Less than any finite distance. Silly! >>>>>>>>> And what is the smallest finite distance? >>>>>>>>> >>>>>>> Note question not answered!! But the correct answer of zero would have >>>>>>> blown TO's argument to blazes, so one can see why he would not care to >>>>>>> answer it. >>>>>> I was away a couple days, but I answered this, not paying attention to >>>>>> that apparent contradiction, since I don't consider 0 really a finite >>>>>> number at all. It's a point with no measure, as every number is measured >>>>>> relative to that point. >>>>> If zero is not a number, how does TO keep the positive numbers separated >>>>> from negatives? >>>> It's not a finite number. It's the origin. A finite number is a finite >>>> distance from the origin. The origin is no distance from itself. >>> Then TO's set of real numbers is two sets separated by a non-number? >>> >>> That does not match anyone else's set of reals. So TO casts himself >>> again into outer darkness re res mathematical. >> It's a 1-D continuum with an origin, a metric space. > > But where are the values of that metric if zero is not one of them? 0 is the origin, the reference point. Measure is difference from 0. 0 is not different from 0. 0 requires no measure. 0 exists. - Tony Orlow (c) 2006 >>>>>>>> When you claim that there are ordinals greater than any finite >>>>>>>> ordinal, >>>>>>>> are you obligated to name the largest finite ordinal? >>>>>>> When you claim there is a LUB to the reals strictly between 0 and 1, >>>>>>> are you required to name the largest real strictly between 0 and 1? >>>>>> No. That's my point. Why should I name the smallest object which is not >>>>>> infinitesimal? >>>>> That is not at all what I asked. So TO is doing his STRAW MAN fallacy >>>>> thing again. >>>> You have no clue what the line of discussion was at this point, do you? >>> I have no idea what TO is talking about, and am reasonably sure he >>> doesn't either. >> Then don't make yourself look silly defending questions and comments >> that are irrelevant. >> >>>>>>> A "LUB" of the naturals does not have to be a natural any more than the >>>>>>> LUB of the the reals strictly between 0 and 1 has to be a real >>>>>>> strictly >>>>>>> between 0 and 1. >>>>>> If it's a discrete set, then I disagree. >>>>> The set {(n-1)/n: n in N} is a discrete set with a LUB which is not a >>>>> member of the set. In fact every strictly increasing sequence having a >>>>> LUB has a LUB which is not a member of the sequence. >>>> That is not "the reals strictly between 0 and 1" but a subset thereof. >>> So there is still no element within either set which is its LUB. >> If the Finlayson reals are used, indeed the LUB is the maximal member of >> the set of reals in [0,1). Ross, is that correct? > > TO appealing to Ross is the blind asking for a lead from the blind. >> Tony I have better hearing, and Ross knows many smells, and we both speak many languages. We're not the one's in the cave, or "garden" as you like to call it. The guano's blooming. Better go pick some. |