An uncountable countable set
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From: Virgil on 23 Oct 2006 21:23 In article <453d5759(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > That function does not have a limit at noon, so cannot be continuous > > there, but it can have a value at noon, and that value can be 0 without > > conflicting with any of the rules of the GE but cannot be anything else > > without conflicting with the rules of the GE. > > Look, fine, relax. Call it a constant discrete increment within the set. > For each iteration, the increase in set size is exactly 9 balls. That > does not change. Every point of the graph of events, where x is > iterations and y is balls in the vase, lies on the line y=9x. The discontinuous function y = 9n is irrelevant, for, among other things, the points (x, y) = (-1/n, 9n) lie on a curve which does not touch the vertical axis, x = 0. For every ball inserted into the vase, there is a time before noon at which it is removed. So that by noon every ball has ben removed. TO can squeal as much as he like, but none of his squealing will change the requirement that for ever ball inserted into the vase there is a time before noon at which it is removed from the vase.
From: Virgil on 23 Oct 2006 21:24 In article <453d577b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453d16f3(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> It has absolutely nothing to do with "cardinality" that I can see. He > >> defines an infinite element of *N as being "larger than any finite > >> number", such that x e N ^ y e *N and ~y e N -> y>x. N is a subset of *N. > > > > But none of Robinson's non-standard numbers are cardinalities. > > No kidding. They actually make sense. Not as cardinalities, they don't.
From: Ross A. Finlayson on 23 Oct 2006 21:43 Virgil wrote: > In article <453d577b(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > Virgil wrote: > > > In article <453d16f3(a)news2.lightlink.com>, > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > > >> It has absolutely nothing to do with "cardinality" that I can see. He > > >> defines an infinite element of *N as being "larger than any finite > > >> number", such that x e N ^ y e *N and ~y e N -> y>x. N is a subset of *N. > > > > > > But none of Robinson's non-standard numbers are cardinalities. > > > > No kidding. They actually make sense. > > Not as cardinalities, they don't. Ah, the point. Virgil, that's not so anyways. Cardinals: ordinals. Yes I'm aware of the difference. What do I say, uh: cardinals are gross qualitative measures. EF coexists with calculus. Ross
From: Tony Orlow on 24 Oct 2006 00:23 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> cbrown(a)cbrownsystems.com wrote: >>>>> Tony Orlow wrote: >>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>>>> Tony Orlow wrote: >>>>>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>>>>>> Tony Orlow wrote: > > <snip> > >>>>> I attempted to describe those assumptions in my previoius post. Did you >>>>> read those assumptions? If so, do you agree with those assumptions? >>>> At this point I don't recall your previous post. I've been off a bit. >>> Well, allow me to repeat them here (with two minor changes): >>> >>> In order to interpret the problem >>> >>> "At each time t = -1/n where n is a (strictly positive) natural number, >>> we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the >>> vase, and remove the ball labelled n from the vase. What is the number >>> of balls in the vase at time t=0?" >>> >>> I make the following simple (and I would claim, fairly uncontroversial >>> and natural) assumptions: >>> >>> --- (object permanence) >>> >>> (1) When we speak of a time t, we mean some real number t. >>> >>> (2) If a ball is in the vase at any time t0, there is a time t <= t0 >>> for which we can say "that ball was placed in the vase at time t". >>> >>> (3) If a ball is placed in the vase at time t1 and it is not removed >>> from the vase at some time t where t1 <= t <= t2, then that ball is in >>> the vase at time t2. >>> >>> (4) If a ball is removed from the vase at time t1, and there is no time >>> t such that t1 < t <= t2 when that ball is placed in the vase, then >>> that ball is not in the vase at time t2. >>> >>> ---- (obedience to the problem constraints) >>> >>> (5) If a ball is placed in the vase at some time t, it must be in >>> accordance with the description given in the problem: it must be a ball >>> with a natural number n on it, and the time t at which it is placed in >>> the vase must be -1/floor(n/10). >>> >>> (6) If a ball is removed from the vase at some time t, it must be in >>> accordance with the description given in the problem: it must be a ball >>> with a natural number n on it, and the time t at which it is removed >>> from the vase must be -1/n. >>> >>> (7) If n is a natural number with n > 0, then the ball labelled n is >>> placed in the vase at some time t1; and it is removed from the vase at >>> some time t2. >>> >>> --- (very general definition of "the vase is empty at noon") >>> >>> (8) the number of balls in the vase at time t=0 is 0 if, and only if, >>> the statement "there is a ball in the vase at time t=0" is false. >>> >>> --- >>> >>> Perhaps you would add other assumptions (9), (10), etc.; but my >>> question is: >>> >>> Given the problem statement, do you agree that /each/ of these >>> assumptions, /on its own/, is reasonable and not just some arbitrary >>> statement plucked out of thin air? >>> >>> If not, which assumption(s) is(are) not reasonable or is(are) >>> unneccessarily arbitrary? >>> >>> <snip> >> Those all look reasonable to me as I read them. > > Good. I propose that if we agree that these statements are true, then > we also agree that any logical implications that follow from, these > statements we also agree to be true. > >> I don't see any >> statement regarding the fact that ten balls are added for every one >> removed, though that can be surmised from the insertion and removal >> schedule. > > It certainly can be deduced from (1)..(8). But it is worth noting that > it was already the case that the statement "for /every/ 10 balls added, > 1 ball is removed" is not /explicitly/ stated at any point the original > problem statement. Uh, yes, it is. It is stated that the first 10 balls not so far inserted will be inserted, then the lowest numbered ball removed. No more balls will be removed until we repeat this step, inserting 10 more balls. That's explicitly stated. > > It is /already/ something we had to deduce - for example, by the > (trivial, but neccessary) logical argument that if t is a time when any > balls are added, then t is a time that 10 balls are added and also a > time when a ball is removed, That's one circuitous way to prove what was stated from the beginning. > > But we should be careful regarding what we are really claiming in this > case. Equally, the statement "for every ball that is added, 1 ball is > removed" can also be deduced from (7), (5) and (6) (in this case we > don't require that the adding and removing occur at the same time t). > Isn't time considered, by you and others, to be a crucial element of the argument? I mean, for my part, it's enough to say we have some infinite sequence of +10-1, or +9, and that it diverges to an infinite value. That's clear. But, if the set-theoretical argument rests on time, doesn't it matter whether you preserve the order of events in time? Are you allowed to change the problem with imagined discontinuities and time vortexes, rearranging events in order to get whatever answer you want? Sure, you can "prove" that kind of stuff from "the axioms", but not while preserving the stated order of events. >> That's the salient fact here. > > Regardless of its saliency, we can at least agree that it follows > logically from (1)-(8). Not without changing the experiment. You agree that it's reasonable to conclude that every time a ball is removed 10 are added. You then suggest that, if we rearrange events by making them happen at different times than originally scheduled, we can argue that the numbers in and out are equal. The point is, you can't do that, and at the same time claim to be discussing the original problem. Can you? > >> You never remove as many as you >> add, so you can't end up empty. > > That is not a conclusion we can draw from (1)..(8). Yes it is, given the times of insertions and removals. The limit is oo at t=0. Instead, this is > the unstated assumption to which you return again and again: > > (Proposition T) If you never remove as many as you add, then there is a > ball in the vase at time 0. If you never remove as many as you add then the vase can only become fuller and is at no time any less full than it was at any time before. > > If we simply /require/ (T) to be true, and da
From: Tony Orlow on 24 Oct 2006 00:42
David R Tribble wrote: > David R Tribble wrote: >>> [Your H-riffics] 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". > > Tony Orlow wrote: >>> 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. > > David R Tribble wrote: >>> Your definition of your H-riffic numbers excludes unending strings. > > Tony Orlow wrote: >>> Since when? Do the digital reals exclude unending strings? > > David R Tribble wrote: >>> You misunderstand. Your H-riffics are simply finite-length paths >>> (a.k.a. the nodes) of a binary tree. Your definition precludes >>> infinite-length paths as H-riffic numbers. > > Tony Orlow wrote: >> What part of my definition says that? For the positives: >> >> 1 e H >> x e H -> 2^x e H >> x e H -> 2^-x e H > > Exactly. If you list these H-riffic numbers as a binary tree, each one > is a node in the tree along a finite-length path. > David, I think I figuredout what your confusion is on this. I presented the H-riffics in "Well Ordering the Reals" as a proposed well ordering. As it turns out, indeed, it would seem that the well ordered version of this set would have to be countable, as you say, much like the rationals, or rather, the finite length decimal reals. But, well ordering be damned, if infinite-length strings are allowed in the H-riffics, it covers the uncountable reals. > > David R Tribble wrote: >>> So 3 can't be a valid H-riffic, and neither can any of its successors. > > Tony Orlow wrote: >>> Nice fantasy, but that's all it is. I suppose 1/3 doesn't exist in >>> decimal either. > > David R Tribble wrote: >>> As I said, you misunderstand. Please demonstrate how 3 (or any >>> multiple or power of 3, for that matter) meets your defintion of an >>> H-riffic number. You claim it (they) do, and I'm asking you for proof. > > Tony Orlow wrote: >> That's something I have to get back to, I suppose, but Dave Rusin had >> confirmed that a base-2 H-riffic representation of 3 was a repeating >> string, much like 1/3 in decimal. It was something like 2^-2^2^-2... > > Exactly. Which means it is not a node in the binary tree of H-riffics. > So it's not an H-riffic number. > Consider the binary reals as a binary tree. 1/3 is .11111..... It's a real, with an unending path. Each real IS an unending path in the tree. The terminating binary reals simply end with an unending string of 0's, so they can be identified either with the node which begins that string of 0's, or with the entire unending path. For consistency with those which have no least significant 1 bit, all such reals should be considered, not nodes, but unending paths. The same goes for the H-riffics. When you have, say, 2^-1/2 as an H-riffic, that's 2^-2^-1, and 1 is 2^0, and 0 is 2^-oo, and oo, of course, is oo^oo. So, 1/sqrt(2) can be represented as 2^-2^-2^2-oo^oo^oo^oo.... in H-riffic notation as 0110100000.... It's a terminating H-riffic. 3 isn't. > > David R Tribble wrote: >>> I know you don't get this, but go back and read your own definition. >>> Every H-riffic corresponds to a node in an infinite, but countable, >>> binary tree. > > Tony Orlow wrote: >>> No, like the reals, it corresponds to a path in the tree. > > David R Tribble wrote: >>> No, read your own definition again. Each H-riffic is a finite node >>> along a path in a binary tree. > >> I'm not sure which definition of an H-riffic you're referring to. Are >> you sure you're not talking about the T-riffics? That's a countably >> infinite set of strings, each being finite in length but representing >> infinite values. Not all infinite values can be represented, since they >> rely on infinite repeating strings between countable neighborhoods, >> making the set countable. Is that what you're talking about? :) > > No. See above. > Where does it say anything about a node in my definition, or whether strings can be infinite? Your baseless declarations about my definitions don't fly. > > David R Tribble wrote: >>> The H-riffics is only a countable subset of the reals, and omits an >>> uncountable number of reals. > > Tony Orlow wrote: >>> Just like all finite-length reals. That is only a countable set. > > David R Tribble wrote: >>> Exactly. The H-riffics exclude an uncountable number of reals, >>> and thus do not cover all the reals. > > Tony Orlow wrote: >> What makes you think infinite-length strings are excluded? They're not, >> in either of my riffic number systems. > > You're confused. Infinite-length fractions are not excluded, > obviously. But we're not talking about fractions, we're talking about > each H-riffic being a node in the binary tree that lists all of them. > Each H-riffic is a node on a finite-length path in the tree. Who the hell said that? Is this your number system now, that you get to declare that my H-riffics are nodes in your tree? Get real. > > Which is why 3 (or any multiple or power of 3) is not an H-riffic. > Nor are most reals, for exactly the same reason. For reasons that you make up out of the blue. > > Assume that 3 is an H-riffic, a node at the "end" of an infinite > length path in the tree. Is that "last" fork a left or a right fork > (i.e., a 2^x or a 2^-x fork)? And at what node would the successor > to 3 be on? > Who said it was a node? Not me. |