An uncountable countable set
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From: Tony Orlow on 3 Oct 2006 02:17 David R Tribble wrote: > Tony Orlow wrote: >>> For the sake of this argument, we can talk about infinite reals, of >>> which infinite whole numbers are a subset. > > David R Tribble wrote: >>> What are these "infinite reals" and "infinite whole numbers" that you >>> speak of so much? > > Tony Orlow wrote: >>> The very same, with no restriction of finiteness. Any T-riffic number >>> has successor. :) > > David R Tribble wrote: >>> It's sporting of you to drop the requirement that all the naturals in N >>> have to be finite, but since all of them are, it's meaningless to say >>> "with no restriction of finiteness". That's kind of like saying N >>> contains all naturals "with no restriction of non-integer values". >>> I can say that, but it does not change the fact that all the members >>> of N are integers. > > Tony Orlow wrote: >> Is the successor to ...11110000 not equal to ...11110001? > > I don't know - how are you defining those numbers? > > Even if it is (or is not), what does it have to do with the finite > naturals? Are you saying that those "infinite naturals" are > somehow successors to the finite naturals? How? > > > David R Tribble wrote: >>> So I ask again, where are those infinite naturals and reals you keep >>> talking about? It's obvious they are not in N. > > Tony Orlow wrote: >> [No] it's not. > > Every member of N has a finite successor. Can you prove that your > "infinite naturals" are members of N? > Yes, if "finite successor" is the only criterion. To prove finiteness of such a string: The bits over each sequence are indexed by natural numbers, which are all finite, yes? For any finite bit position, the string up to and including that bit position can only represent a finite value, yes? Therefore, there is no bit position where the string can have represented anything but a finite value, see? If the length is potentially, but not actually, infinite, so with the value. To prove successorship of such a string: The rule for successorship for finite values is 1. Find the rightmost (least significant) 0 2. Invert from that 0 rightwards This works for all values where there is a rightmost 0. That excludes ....111, which can only have successor given ignored overflow, allowable in some cases. You don't really question why the successor to ...11110000 is equal to ....11110001, do you? TOny
From: Tony Orlow on 3 Oct 2006 02:18 David R Tribble wrote: > Tony Orlow schrieb: >>> Do I "misunderstand" that if you remove balls 1, then 11, then >>> 21, etc, that the vase will NOT be empty? > > mueckenh wrote: >>> This is an extremely good example that shows that set theory is at >>> least for physics and, more generally, for any science, completely >>> meaningless. Because the numbers on the balls cannot play any role >>> except for set-theory-believers. > > Tony Orlow wrote: >> Yes. I was flabbergasted by this example of "logic". The amazing thing >> is, set theory is supposed to apply where we know nothing except for the >> membership status of each element in a set, and yet, here is applied >> this property of labels that set theorists claim is crucial to answering >> the question. > > How do you know the membership status of each element if you > have no way to distinguish the elements from each other? > That's a very good question.
From: Tony Orlow on 3 Oct 2006 02:27 David R Tribble wrote: > Tony Orlow wrote: >>> You drew that from my suggestion of the number circle, and that ...11111 >>> could be considered equal to -1. Since then, I looked it up. I'm not the >>> first to think that. It's one of two perspectives on the number line. >>> It's either really straight, or circular with infinite radius, making it >>> infinitesimally straight. The latter describes the finite universe, and >>> the former, the limit. But, you knew that, and are just trying to have fun. > > David R Tribble wrote: >>> You are drawing geometric conclusions that are not warranted. >>> The Projective Real Line is simply R U {oo}. Adding unsigned oo >>> to the set allows certain arithmetic operations to be performed >>> that are undefined in the regular real set. >>> >>> But simply adding the limit point oo to the set does not actually make >>> it a "circle", because oo has no predecessor or successor, and >>> certain operations like oo+1 and oo+oo are still meaningless within >>> the set. > > Tony Orlow wrote: >> You do realize that my statements involve a considerable amount of >> personal reflection, don't you? There is more to the number circle than >> "proven". In the binary number circle, "100...000" is both positive and >> negative infinity. > > By what axioms? > By the theorem that the negative of a 2's complement number is calculated as the bitwise inverse plus 1. It is the largest value, that is, furthest from 0 in the additive circle, and also its own additive inverse. What says this to you? Inverse of 0? oo? > > David R Tribble wrote: >>> See: >>> http://en.wikipedia.org/wiki/Projective_line#Real_projective_line >>> http://en.wikipedia.org/wiki/Division_by_zero#Real_projective_line >>> http://en.wikipedia.org/wiki/Extended_real_number_line >>> >>> You obviously have something else in mind when you talk about >>> the "number circle". Perhaps you could actually define it some time? > > Tony Orlow wrote: >> There are a number of concepts in this area. Get acquainted with those >> pages, think, and come back and talk. > > And that will allow you to define the terms of your number theory? > > You underestimate the degree to which we understand your theories. Or, perhaps you overestimate that degree. > It's pretty obvious that the mathematically inclined among us follow > what you're saying, because, honestly, it's really not that complicated > or profound. Most of it also happens to be self-contradictory, which > is where most of us have a problem "seeing" it. Which makes it > incumbent upon you to define your ideas a little more clearly and > a lot more formally. If you see a contradiction, please say what it is. I'm happy to discuss whatever seems wrong. I don't think it's self-contradictory. I think it's other-contradictory. And if it's that basic (it's not meant to be complicated or profound) then why isn't it mainstream? > > It's not a failing on my part that your ideas are not consistent, but > rather a failing on your part not to recognize the logical conclusions > that prove these inconsistencies. > What inconsistencies?
From: Tony Orlow on 3 Oct 2006 02:31 Virgil wrote: > In article <452160ef(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Randy Poe wrote: > >>> You're right. So it's back to Tony. The quote is yours (Tony), so >>> just follow the attributes back to "if noon exists, that's >>> when the vase empties", and that's when you (Tony) >>> said such a thing. >>> >>> - Randy >>> >> Okay Randy, but does it, or not? Is it non-empty, then empty? Does this >> change in state occur before noon? After noon? Or, at noon? > > It is at a particular state at any particular time, but as the states do > not form a continuum, one gets jump discontinuities in the number of > balls in the vase in a neighborhood of each relevant time. And in every > neighborhood of noon, there are infinitely many such discontinuities. Yo! In da House! Diss! Diss! Diss! Discontinuities! This is the problem with the ordinals. They're not "real". There is no "line". Time to let von Neumann's snow job slide. Global warming and all. Toodles. Tony
From: Tony Orlow on 3 Oct 2006 02:35
Virgil wrote: > In article <45216233(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David R Tribble wrote: >>> Virgil wrote: >>>>> Except for the first 10 balls, each insertion follow a removal and with >>>>> no exceptions each removal follows an insertion. >>> Tony Orlow wrote: >>>> Which is why you have to have -9 balls at some point, so you can add 10, >>>> remove 1, and have an empty vase. >>> "At some point". Is that at the last moment before noon, when the >>> last 10 balls are added to the vase? >>> >> Yes, at the end of the previous iteration. If the vase is to become >> empty, it must be according to the rules of the gedanken. > > But the "rules of the gedanken" specifically forbid any "last 10 balls", > by specifying an ENDLESS sequence of 10 ball additions. No, the gedanken starts with adding ten balls, then removing one, then repeating forever. The gedanken does not predict its own conclusion. The question is whether the vase can empty. It can't, as long as more balls are coming in than going out. That's clear to the sane. This theory is insane, if an empty vase is what it predicts. I like your shoes. Tony |