An uncountable countable set
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From: Virgil on 12 Oct 2006 13:59 In article <452e55b6(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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 So that TO will insist that although every label has been removed from the vase and every ball is labelled, some of the balls remain?
From: David Marcus on 12 Oct 2006 14:01 Ross A. Finlayson wrote: > 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. Please give the precise statement of the problem in English that you are using and also your translation of the problem into Mathematics. -- David Marcus
From: Virgil on 12 Oct 2006 14:04 In article <452e55e7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452d8ef0(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> 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. > > > > The labeling is as relevant in those cases as in the case of two balls, > > numbered one and two, resepectively, with ball two being put in and left > > there and ball one not being put in at all. > > If you put ball 1 instead of 2, does that change the number of balls > left in the vase? It shouldn't. It certainly affects WHICH ball is in the vase, even in this simple case. In the OP's problem in which it is always the lost numbered ball which is removed, one can must allow that for each n in N, the ball labelled n is not in the vase at noon. So what balls does TO claim ARE in the vase at noon?
From: David Marcus on 12 Oct 2006 14:05 mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > In article <1160310643.181133.6720(a)b28g2000cwb.googlegroups.com> mueckenh= > @rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > AXIOM OF INFINITY Vla There exists at least one set Z with the > > > following properties: > > > (i) O eps Z > > > (ii) if x eps Z, also {x} eps Z. > > > > > > There are several verbal formulations dispersed over the literature > > > without any "all". > > > > Perhaps. Does this mean that there are some x in Z such that {x} not in > > Z? Or, what do you mean? > > I did not interpret what it may mean. I said that in verbal forms of > the axioms the word "all" does not appear. I said: I never came across > the word "all" in connection with this axiom. > > > > > In German: Unendlichkeitsaxiom: Es gibt eine Menge, > > > die die leere Menge enth=E4lt, und wenn sie die Menge A enth=E4lt, so > > > enth=E4lt sie auch die Menge A U {A} (oder die Menge {A}). > > > > Do you not think that that means that for all A in that set, also {A} in > > that set? If not, why not? > > I wrote: > > > The axiom of infinity does only state n+1 exists if n is given. It is > > > realized by he numbers of my list. > You replied: > > That is *not* what the axiom of infinity states. The axiom states that > > there exists a set that contains all the successors of its elements. > And I reply: > That is wrong. The axiom says: For all possible n in all possible > cases: If n is a natural number, then n+1 is a natural number too. It > does *not* say that it is meaningful to speak of *all* natural numbers. > It does not say that the set N does actually or completely exists. It > does not say that this infinity is finished, i.e., that all natural > numbers can exist together. Well: "there is a set". But the meaning of > these three words depends heavily on the interpretation. Therefore, the > interpretation which usually is given cannot be free of arbitrariness, > and it leads to contradictions. That is the "solid" foundation of > present matheology. You wrote that standard mathematics "leads to contradictions". Do you mean that standard mathematics can prove a statement P and also prove the statement "not P"? If so, please give the statement and the proofs. -- David Marcus
From: Virgil on 12 Oct 2006 14:06
In article <452e578c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> David Marcus wrote: > >>>>>>>>> Tony Orlow 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. > >>>>>>>>>>> David R Tribble wrote: > >>>>>>>>>>>>> "At some point". Is that at the last moment before noon, when > >>>>>>>>>>>>> the > >>>>>>>>>>>>> last 10 balls are added to the vase? > >>>>>>>>>>>>> > >>>>>>>>>>> Tony Orlow wrote: > >>>>>>>>>>>> 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. > >>>>>>>>>>> The rules don't mention a last moment. > >>>>>>>>>>> > >>>>>>>>>> The conclusion you come to is that the vase empties. As balls are > >>>>>>>>>> removed one at a time, that implies there is a last ball removed, > >>>>>>>>>> does > >>>>>>>>>> it not? > >>>>>>>>> Please state the problem in English ("vase", "balls", "time", > >>>>>>>>> "remove") > >>>>>>>>> and also state your translation of the problem into Mathematics > >>>>>>>>> (sets, > >>>>>>>>> functions, numbers). > >>>>>>>> Given an unfillable vase and an infinite set of balls, we are to > >>>>>>>> insert > >>>>>>>> 10 balls in the vase, remove 1, and repeat indefinitely. In order to > >>>>>>>> have a definite conclusion to this experiment in infinity, we will > >>>>>>>> perform the first iteration at a minute before noon, the next at a > >>>>>>>> half > >>>>>>>> minute before noon, etc, so that iteration n (starting at 0) occurs > >>>>>>>> at > >>>>>>>> noon-1/2^n) minutes, and the infinite sequence is done at noon. The > >>>>>>>> question is, what will we find in the vase at noon? > >>>>>>> OK. That is the English version. Now, what is the translation into > >>>>>>> Mathematics? > >>>>>> Can you only eat a crumb at a time? I gave you the infinite series > >>>>>> interpretation of the problem in that paragraph, right after you > >>>>>> snipped. Perhaps you should comment after each entire paragraph, or > >>>>>> after reading the entire post. I'm not much into answering the same > >>>>>> question multiple times per person. > >>>>> I snipped it because it wasn't a statement of the problem, as far as I > >>>>> could see, but rather various conclusions that one might draw. > >>>> I drew those conclusions from the statement of the problem, with and > >>>> without the labels. > >>> 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). > > > > So, you are saying that the problem translates into the following sum: > > > > sum_{i=1}^infty (10-1). > > > > But, when you translate a problem into mathematics, you need to say how > > each part of the problem is represented in the mathematics. In > > particular, how are the times represented in your model and what in your > > model represents the number of balls in the vase? > > > > Time is actually irrelevant. The sequence is measured in iterations as > n->oo, and the number of balls in the vase at iteration n is represented > by sum(x=1->n: 9). The limit of this sum as x diverges also diverges in > linear fashion. But that divergence is irrelevant in the face of the question "for any n in N, is ball n in the vase at noon?" |