An uncountable countable set
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From: Tony Orlow on 12 Oct 2006 22:20 Mike Kelly wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Tony Orlow wrote: >>>> cbrown(a)cbrownsystems.com wrote: >>>>> Tony Orlow wrote: >>>>>> Virgil wrote: >>>>>>> In article <452d11ca(a)news2.lightlink.com>, >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>> >>>>>>>>> I'm sorry, but I can't separate your statement of the problem from your >>>>>>>>> conclusions. Please give just the statement. >>>>>>>>> >>>>>>>> The sequence of events consists of adding 10 and removing 1, an infinite >>>>>>>> number of times. In other words, it's an infinite series of (+10-1). >>>>>>> That deliberately and specifically omits the requirement of identifying >>>>>>> and tracking each ball individually as required in the originally stated >>>>>>> problem, in which each ball is uniquely identified and tracked. >>>>>> The original statement contrasted two situations which both matched this >>>>>> scenario. The difference between them was the label on the ball removed >>>>>> at each iteration, and yet, that's not relevant to how many balls are in >>>>>> the vase at, or before, noon. >>>>> Do you think that the numbering of the balls is not relevant to >>>>> determining the answer to the question "Is there a ball labelled 15 in >>>>> the vase at 1/20 second before midnight?" >>>>> >>>>> Cheers - Chas >>>>> >>>> If it's a question specifically about the labels, as that is, then it's >>>> relevant. It's not relevant to the number of balls in the vase at any >>>> time, as long as the sequence of inserting 10 and removing 1 is the same. >>>> >>>> Tony >>> Ah, but noon is not a part of the sequence of iterations. No more than >>> 0 is an element of the sequence 1, 1/2, 1/4, 1/8, .... >>> >>> The question asks how many balls are in the vase at noon. Not at some >>> iteration. >>> >> Ah, but if noon is not part of the sequence, then nothing from the >> sequence has anything whatsoever to do with how many balls are in the >> vase at noon. I think there are three, you know, the number of licks it >> takes to get to the tootsie roll center of a tootsie pop. That makes >> about as much sense as saying an infinite number of them vanish. If noon >> is not part of your sequence, then it's a nonsensical question, and if >> it is, then the limit applies. > > So, do you think 0 is an element of the sequence 1, 1/2, 1/4, 1/8, ... > ? > No, Iv'v come around. I agree with you now. :) 0 has nothing whatsoever to do with that sequence at all. Isn't that right?
From: Ross A. Finlayson on 12 Oct 2006 22:41 Michael Stemper wrote: > In article <1160613324.049353.270480(a)k70g2000cwa.googlegroups.com>, Ross A. Finlayson writes: > >David Marcus wrote: > >> Ross A. Finlayson wrote: > >> > David Marcus wrote: > > >> > > So, you aren't saying that ZF is inconsistent. You are saying that you > >> > > prefer to use a different system of axioms. Is that correct? > > >> > No, David, Dave, I say ZF is inconsistent. That doesn't mean all its > >> > results are false. It just lets me say whatever I want about a less > >> > inconsistent set of "axioms". > > >> So, you are saying that ZF is inconsistent. By "inconsistent", do you > >> mean that ZF proves both P and not P, for some statement P? If so, > >> please be explicit: what is the statement and what is the proof in ZF of > >> it? > > >Build a set: {x: true}, it's a set, > > No, "{x: true}" is gibberish. Just putting down a couple of braces and > throwing random text inside them does not generate a set. At least, no > according to the axioms of ZF that you're trying to refute. > > -- > Michael F. Stemper > #include <Standard_Disclaimer> > There is three erors in this sentence. Hi, Regularity, the axiom of regularity also called foundation in the spirit if not always letter of well-foundedness, re RRA, regularity is an axiom that says sets like the Universe don't exist. So, I say there is a universe or there is no universal quantification, and, ZF has no universe, so, ZF has no quantifier, illustrating how that leads to systemic inconsisency, thus that as I described in the prequel ZF contradicts itself, yes, invalidating all of ZF's own statements, by itself. That notation was casual. How about "for any x: x E X", and that's not "for any x E X...". Sometimes people talk about axioms in terms of their truth or falseness, falsity, for example asserting that the universe exists, i.e. V = L, the universe is the constructible universe. A lot of great things happen when V = L. That's why some people think it's a "true" axiom. How's coding? Ross
From: Ross A. Finlayson on 12 Oct 2006 22:57 Tony Orlow wrote: > Ross A. Finlayson wrote: > > Randy Poe wrote: > >> Tony Orlow wrote: > >>> David Marcus wrote: > >>>> Virgil wrote: > >>>>> In article <452d11ca(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>>> I'm sorry, but I can't separate your statement of the problem from your > >>>>>>> conclusions. Please give just the statement. > >>>>>> The sequence of events consists of adding 10 and removing 1, an infinite > >>>>>> number of times. In other words, it's an infinite series of (+10-1). > >>>>> That deliberately and specifically omits the requirement of identifying > >>>>> and tracking each ball individually as required in the originally stated > >>>>> problem, in which each ball is uniquely identified and tracked. > >>>> It would seem best to include the ball ID numbers in the model. > >>>> > >>> Changing the label on a ball does not make it any less of a ball, and > >>> won't make it disappear. If I put 8 balls in an empty vase, and remove > >>> 4, you know there are 4 remaining, and it would be insane to claim that > >>> you could not solve that problem without knowing the names of the balls > >>> individually. > >> That's a red herring. It's not the name of the ball that's relevant, > >> but whether for any particular ball it is or isn't removed. > >> > >>> Likewise, adding labels to the balls in this infinite case > >>> does not add any information as far as the quantity of balls. > >> No, but what the labels do is let us talk about a particular > >> ball, to answer the question "is this ball removed"? > >> > >> If there is a ball which is not removed, whatever label > >> is applied to it, then it is still in the vase. > >> > >> If there is a ball which is removed, whatever label is > >> applied to it, then it is not in the vase. > >> > >>> That is > >>> entirely covered by the sequence of insertions and removals, quantitatively. > >> Specifically, that for each particular ball (whatever you > >> want to label it), there is a time when it comes out. > >> > >> - Randy > > > > > > I describe some conditions on the ball and vase problem that can help > > make it more realistic. > > > > The golem with the marker in the vase, where you can't reach into the > > vase, if you want one ball out for putting ten in, there would need to > > be infinitely many golems if each can only hold one ball. > > > > Recently in this discussion about infinite sets and so on one of the > > talking points about Cantor that has emerged is that he counts > > backwards from infinity. > > > > The empty-vasers construct the argument that for any ball labelled n, > > where each ball has some factory serial, they can denote some time > > 1/2^n where that number has been retrieved from the vase. By the same > > token, at time 1/2^n, ten balls were just placed in the vase. For each > > of those, the various times they are retrieved from the vase are > > exactly specified, and, at each of those ten more new ones are added to > > the vase. At each constructed time, for n many iterations, the count > > of balls in the vase is 9n. > > > > The count of balls in the vase is the difference of two divergent > > series. > > > > > > Ross > > > > Exactly, though I dunno about the Golems. Well, you see Tony, you put a Maxwell demon in the vase, and it brings thermodynamics with it, and the vase explodes. Then guys are standing around going hell why did I put a Maxwell demon in the vase that just ruined the entire experiment. Keeping track of the balls, the Maxwell demon's entropy would increase to the point where it would disrupt. These little golems are much simpler and have a lot less baggage. To speak candidly, then you can put them in every "thought" experiment. Ross -- Ah, Gran Turismo, I miss you so.
From: Virgil on 12 Oct 2006 23:17 In article <452ef2bc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> Virgil wrote: > >>>>> In article <452d11ca(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>>> I'm sorry, but I can't separate your statement of the problem from > >>>>>>> your > >>>>>>> conclusions. Please give just the statement. > >>>>>>> > >>>>>> The sequence of events consists of adding 10 and removing 1, an > >>>>>> infinite > >>>>>> number of times. In other words, it's an infinite series of (+10-1). > >>>>> That deliberately and specifically omits the requirement of identifying > >>>>> and tracking each ball individually as required in the originally > >>>>> stated > >>>>> problem, in which each ball is uniquely identified and tracked. > >>>> The original statement contrasted two situations which both matched this > >>>> scenario. The difference between them was the label on the ball removed > >>>> at each iteration, and yet, that's not relevant to how many balls are in > >>>> the vase at, or before, noon. > >>> Do you think that the numbering of the balls is not relevant to > >>> determining the answer to the question "Is there a ball labelled 15 in > >>> the vase at 1/20 second before midnight?" > >>> > >>> Cheers - Chas > >>> > >> If it's a question specifically about the labels, as that is, then it's > >> relevant. It's not relevant to the number of balls in the vase at any > >> time, as long as the sequence of inserting 10 and removing 1 is the same. > >> > > > > Putting aside the question of /how/ (limit? sum of binary functions?) > > one determines the /number/ of balls in the vase at time t for a > > moment... > > > > Do you then agree that there is some explicit relationship described in > > the problem between what time it is, and whether any particular > > labelled ball, for example the ball labelled 15, is in the vase at that > > time? > > For any finite time before noon, when iterations of the problem are > temporally distinguishable, yes, but at noon, no. So that TO says that at noon, after a particular ball, like number 15, has been removed, it may not be distinguishable whether ball 15 is in the vase of not? What a weird world TO lives in.
From: Virgil on 12 Oct 2006 23:20
In article <452ef411(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > The whole point of the Zeno machine is to conceive of completing this > infinite series of events, and yet, it compresses the vast majority of > events into a single moment at noon, making it impossible to distinguish > them. Actually, in either version of the original problem, NONE of the transactions take place AT noon. Each of them precedes noon. |