An uncountable countable set
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From: Lester Zick on 21 Oct 2006 13:35 On 20 Oct 2006 13:26:08 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Tony Orlow wrote: >> I'm reading Non-Standard Analysis instead. > >What book? > >You really would be better served by having your basics in set theory >and mathematical logic in order first and then taking on non-standard >analysis. Why? >> Robinson agrees there's no >> smallest infinity, > >Then that is not the same as the ordering of the ordinals we're talking >about. I very much doubt that Robinson claims that there is not a least >infinite ordinal. > >Please tell me exactly what passages or theorems you are referring to >in Robinson's work so that I can see exactly what it is you are talking >about. > >MoeBlee ~v~~
From: Lester Zick on 21 Oct 2006 13:40 On 20 Oct 2006 11:55:38 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Han de Bruijn wrote: >> > Anyway, what is confusing about the theorem that there exist finite >> > ordinals and that there exist infinite ordinals? > >What do you find confusing about the following statement: > >There exists an x such that the following hold: x is well ordered by >membership; x is transitive by membership; there is no natural number >that is bijectable with x. What I personally find confusing is that you seem to think the foregoing says something that needs saying. ~v~~
From: Lester Zick on 21 Oct 2006 14:06 On 20 Oct 2006 13:39:32 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >MoeBlee wrote: >> Tony Orlow wrote: >> > I'm reading Non-Standard Analysis instead. >> >> What book? >> >> You really would be better served by having your basics in set theory >> and mathematical logic in order first and then taking on non-standard >> analysis. >> >> > Robinson agrees there's no >> > smallest infinity, >> >> Then that is not the same as the ordering of the ordinals we're talking >> about. I very much doubt that Robinson claims that there is not a least >> infinite ordinal. >> >> Please tell me exactly what passages or theorems you are referring to >> in Robinson's work so that I can see exactly what it is you are talking >> about. > >And this reminds me that you never did come to understand the >difference between cardinality and ordering. I and others have pointed >out to you that you conflate these. One doesn't even need non-standard >analysis to provide an ordering in which there is a set S with no least >member yet with every member of S greater than some set with no >greatest member. But that ordering is not an ordering by cardinality. >Yes, PA has models in which there are different orderings so that >objects are called 'infinite' per these orderings, but this is NOT the >same sense of 'infinite' as that of the cardinality sense. And we can >define a division operation on these "infinite" objects to get >infinitesimals, but again, this is not the same as cardinality. > >Moreover, you must be very careful to distinguish between the proof of >the existence of certain models and a RECURSIVE axiomatization for a >theory of which the model is a model of. Ah another lecture in neomathspeak. See the problem is that Tony just doesn't agree with you. And there is nothing in what you say which can bridge that gap because nothing in the orthodox catechism of standard set methodology (I'm assuming it's standard) can be demonstrated true. You and he just have different perspectives on the problem and nothing in what you have to say has any relevance to what Tony believes any more than what Tony believes has any relevance to what you believe. Which is undoubtedly why he has begun to study non standard analysis before satisfying your criteria for the standard set methodologies you believe in. And though your beliefs are pervasive, to the extent they aren't demonstrably true your arguments are irrelevant because they're unable to effect a suspension of disbelief in Tony. In general terms it's called a crisis of faith. ~v~~
From: cbrown on 21 Oct 2006 22:30 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> > >> You have agreed with everything so far. At every point before noon balls > >> remain. > > > > To be precise, the assertions above all imply that at every time t = > > -1/n, where n is a natural number, there are balls in the vase. > > > > But that *alone* does not even include every time t before noon; let > > alone every time t. For example, notice that nowhere above do you or I > > /explicitly/ assert: "at t=-2/3, the number of balls in the vase is a > > positive finite number". > > > > We assert something specific about t = -2/4, and something specific > > about t = -1/3, but nowhere do we directly state somthing about t = > > -2/3. > > > > On the other hand, given the problem statement, I think we would both > > /agree/ that there "should be" an obvious (perhaps even unique) > > well-defined answer to the question : "what is the number of balls in > > the vase at time t = -1/pi?" > > > > Assuming in the remaining statements that one agrees with the previous > > statement, this leads us to the question: what are the unstated > > assumptons that allow to agree that this must be the case? > > > > 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> > > > >> You claim nothing changes at noon. > > > > Where, exactly, above do I claim that "nothing changes at noon"? > > Do you disagree with the other standard-bearers, and claim that > something DOES occur at noon? That is not a response to /my/ question "where, exactly, above do I claim that 'nothing changes at noon'?" /I/ don't claim that "something occurs at noon"; nor do /I/ claim that "nothing occurs at noon". /YOU/ are claiming that the truth of these statements follows logically from our assumptions; but until you make some kind of mathematical statement which corresponds to "something occurs at noon", I really can't address it as a /mathematical/ question. <snip> > >>>>> Do you accept the above statements, or do you still claim that there is > >>>>> /no/ valid proof that ball 15 is not in the vase at t=0? > > I said that any specific ball was obviously out of the vase at noon. > That's good: we at least agree that it logically follows from (1) - (8) that there are no labelled balls in the vase at t=0. What I honestly find baffling is your repeated claim that it doesn't then logically follow from assumptions (2) and (5), that if a ball is in the vase at /any/ time, it is a ball which is labelled with a natural number; and so therefore the above statement is logically equivalent to "there are no balls in the vase at t=0". <snip> > >>> Therefore ball 15 is not in the vase at noon; and nothing you > >>> said above challenges the logic of this conclusion. > >>> > >> Of course not. Ball 15 is gone. > >> > > > > You say here, "of course not", but you previously stated that it is > > "not in my purview" to claim /anything at all/ about the state of > > affairs at noon; because "noon does not occur". Therefore, we can > > conclude that you now retract these statements, is that correct? > > > > The state of affairs with regard to the VASE. Ball 15 is obviously out > of the vase. All finitely numbered balls are out. But not before noon. Assuming "But not before noon" = "But not at any time t < 0"; these statements follow from (1)-(8). > And if the experiment continues UNTIL noon, infinitely-numbered balls > are added. There's no way around that. There's nothing in between. > I don't see how you justify your assertion "if the experiment continues UNTIL noon, infinitely-numbered balls are added", if we only assume (1) - (8) above (in particular, it directly contradicts (5)). So my guess is that you are appealing to some /oth
From: Tony Orlow on 22 Oct 2006 04:42
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. > > David R Tribble wrote: >>> 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". > > 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? > > 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. > 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 > > 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. > > As I said, you misundertsand. 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. > 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... > > 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. > > 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? :) > > 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. > > Exactly. The H-riffics exclude an uncountable number of reals, > and thus do not cover all the reals. > What makes you think infinite-length strings are excluded? They're not, in either of my riffic number systems. > > Tony Orlow wrote: >> So, the digital reals are not the reals? Tell it to Cantor the Diagonal. > > Irrelevant. All of the reals can be written in digital form. But the > reals with non-terminating non-repeating fractions form the uncountable > set of irrationals that comprise most of the reals. > Well, sure, and there are always those elements with such non-finite representations when using any finite alphabet to count any truly infinite set. That's true of the H- and T-riffics, too. |