An uncountable countable set
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From: David Marcus on 23 Oct 2006 18:58 Tony Orlow wrote: > The function y=9x is continuous, even if you're only interested in the > values at integral values of x. What does it mean for a function on the integers to be continuous? -- David Marcus
From: David R Tribble on 23 Oct 2006 19:26 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 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. 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. 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. 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. 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?
From: David R Tribble on 23 Oct 2006 19:27 Tony Orlow wrote: > You have agreed with everything so far. At every point before noon balls > remain. You claim nothing changes at noon. Is there something between > noon and "before noon", when those balls disappeared? If not, then they > must still be in there. Of course there is a "something" between "before noon" and "noon" where each ball disappears. At step n, time 2^-n min before noon, ball n is removed. This happens for every ball, since there is a step n for every ball. The balls are removed, one by one, one at each step, before noon.
From: Tony Orlow on 23 Oct 2006 19:40 Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Randy Poe wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>> >>>>>>>> At every time before noon there are a growing number of balls in the >>>>>>>> vase. The only way to actually remove all naturally numbered balls from >>>>>>>> the vase is to actually reach noon, in which case you have extended the >>>>>>>> experiment and added infinitely-numbered balls to the vase. All >>>>>>>> naturally numbered balls will be gone at that point, but the vase will >>>>>>>> be far from empty. >>>>>>> By "infinitely-numbered", do you mean the ball will have something other >>>>>>> than a natural number written on it? E.g., it will have "infinity" >>>>>>> written on it? >>>>>>> >>>>>> Yes, that is precisely what I mean. If the experiment is continued until >>>>>> noon, >>>>> The clock ticks till noon and beyond. However, the explicitly- >>>>> stated insertion times are all before noon. >>>>> >>>>>> so that all naturally numbered balls are actually removed (for at >>>>>> no finite time before noon is this the case), >>>>> There is no finite time before noon when all balls have >>>>> been removed. >>>>> >>>>> However, any particular ball is removed at a finite >>>>> time before noon. >>>>> >>>>>> then any ball inserted at >>>>>> noon must have a number n such that 1/n=0, >>>>> However, there is no ball inserted at noon. >>>>> >>>>>> which is only the case for >>>>>> infinite n. If the experiment does not go until noon, not all naturaly >>>>>> numbered balls are removed. >>>>> The experiment goes past noon. No ball is inserted at noon, >>>>> or past noon. >>>>> >>>>> - Randy >>>>> >>>> Randy, does it not bother you that no ball is removed at noon, >>> ... I agree with that... >>> >>>> and yet, when every ball is removed before noon, >> I should have said "each"... > > Each ball is removed before noon. Every ball is removed > before noon. All the balls are removed before noon. Given > any ball, that ball is removed before noon. > >>> ... I agree with that... >>> >>>> balls remain in the vase? >>> .. I don't agree with that. >>> >>> When did I ever say balls remain in the vase? Every ball >>> is removed before noon. No balls remain in the vase at >>> noon. >> Do you disagree with the statement that, at every time -1/n, when ball n >> is removed, for every n e N, there remain balls n+1 through 10n, or 9n >> balls, in the vase? > > At every time before noon, there are not only finitely many balls > in the vase, there are still infinitely many balls yet to be put in > the vase. Of course, every one of those balls will be removed > before noon. Without exception. > >>>> How do you explain that? >>> I would certainly have difficulty understanding how the vase >>> could be non-empty at noon, given that every ball in the vase >>> is removed before noon. >> There is no ball, in all of N, for which the vase is empty at its >> departure from the vase. > > Yes. There is no last ball. > > But there is no ball which fails to be removed. > >>> But YOU are the one who says the vase is non-empty at >>> noon. I never said such a thing. I'm certainly not going to >>> defend YOUR illogical position. >> I was saying it is non-empty at every one of the finite times before >> noon where any ball is inserted or removed. Do you argue against THAT >> statement? > > I agree with that statement. For t<0, the vase is non-empty. > > At t=0, the vase is empty. > >>> So you now agree that it makes no sense that the vase >>> could be non-empty at noon? That the vase must, in other >>> words, be empty? >> No, you misread. > > You didn't ask me "does it make sense that the vase is > non-empty at noon"? Ah well, I'll answer that question anyway. > No, it doesn't make sense to me that the vase would be > non-empty at noon. Of course it's empty. > > - Randy > Even though it didn't become empty at noon, nor before... Abracadabra!
From: Lester Zick on 23 Oct 2006 19:44
On Mon, 23 Oct 2006 12:08:52 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >stephen(a)nomail.com wrote: >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> > cbrown(a)cbrownsystems.com wrote: >> >> 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". >> >> > It is rather amazing. The logic seems to be that the limit of the number >> > of balls in the vase as we approach noon is infinity, so the number of >> > balls in the vase at noon must be infinity, but all numbered balls have >> > been removed, therefore the infinity of balls in the vase at noon aren't >> > numbered. It does have a sort of surreal appeal. >> >> With the added surreal twist that the limit of the number >> of unnumbered balls in the vase as we approach noon is 0, >> but the number of unnumbered balls in the vase at noon is >> infinite. :) > >I guess consistency isn't required for surrealism. At least Tony doesn't >draw the standard conclusion that Mathematics is inconsistent. He just >pursues his contradictions wherever they may lead. So what? No more unreasonable than pursuing standard mathematical analytical techniques which produce unreasonable results. I imagine Tony just prefers a different mathematical eschatology than the conventional one. Six of one half dozen of the other unless you're suggesting the standard mathematical set paradigm is actually true. ~v~~ |