An uncountable countable set
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From: Virgil on 8 Oct 2006 22:58 In article <4529ac7e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452949b7(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45286ce5$1(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> David R Tribble wrote: > >>>>> What about: > >>>>> sum{n=0 to oo} (10n+1 + ... + 10n+10) - sum{n=1 to oo} (n) > >>>>> The left half specifies the number of balls added to the vase, and > >>>>> the right half specifies those that are removed. > >>>>> > >>>> Do you mean: > >>>> sum{n=0 to oo} (10) - sum{n=0 to oo} (1)? > >>>> That sounds like what you re describing, and termwise the difference is > >>>> sum(n=0 to oo) (9). That's infinite, eh? > >>> But the sums are not given termwise in the question, but sumwise, so > >>> cannot be calculated termwise in your answer, but must be done sumwise. > >>> > >>> And sumwise they are no different. > >> That's bass-ackwards. The gedanken specifically states that the > >> insertion of 10 and the removal of 1 are coupled as an iteration in the > >> process under discussion. > > > > As we are considering the expression suggested by David Tribble, > > > > <quote> > > What about: > > sum{n=0 to oo} (10n+1 + ... + 10n+10) - sum{n=1 to oo} (n) > > The left half specifies the number of balls added to the vase, and > > the right half specifies those that are removed. > > <\quote> > > > > The original problem is, for the moment, irrelevant. > > > > What David wrote didn't even make sense to me, which is why I > re-expressed it. What does {10n+1+...+10n+10} mean except 100n+55, and > what does that signify? Why is he summing this expression from 0, and > summing n from 1? Are we really summing n, anyway? We're not adding the > numbers on the balls to each other, we're just counting them 10- or 1 or > 9-at-a-time. So, it's not the original problem which is irrelevant, but > this meaningless expression which is irrelevant to the original problem. > > > > >> Between every removal of 1 and the successive > >> removal of 1 is an insertion of 10. So, each term '+10' must be coupled > >> with a term '-1', which together make a '+9' per iteration. Where you > >> violate the condition of the experiment that specifies this coupling of > >> events into an iteration, you create your "discontinuity" at noon, and > >> the monster under your bed. > > > > Let A_n(t) be equal to > > 0 at all times, t, when the nth ball is out of the vase, > > 1 at all times, t, when the nth ball is in the vase, and > > undefined at all times, t, when the nth ball is in transition. > > > > Note that noon is not a time of transition for any ball, though it is a > > cluster point of such times. > > Does time obey the law of trichotomy? Can something occur, not before, > not after, and not at the same time as another event, given that the > events happen instantaneously? In general relativity , two events cannot be said to occur at "the same time"" unless they are also in the same place. > > > > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > > at any non-transition time t.Let A_n(t) be equal to > > 0 at all times, t, when the nth ball is out of the vase, > > 1 at all times, t, when the nth ball is in the vase, and > > undefined at all times, t, when the nth ball is in transition. > > What is "in transition". Can't we consider the addition or removal of a > ball to be instantaneous? Would TO's "occurring instantaneously" mean that it does not take place at any point in the time stream? I don't think so. The function I described is undefined at all such points in time at which there is a transition in position from before that moment to after that moment. > > > > > Note that noon is not a time of transition for any ball, though it is a > > cluster point of such times. > > > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > > at any non-transition time t. > > > > B(t) is clearly defined and finite at every non-transition point, as > > being, essentially, a finite sum at every such non-transition point. > > Correct, a linearly increasing finite value. Not increasing between points of discontinuity nor over any interval of time containing times past noon. > > > > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > > > > > B(t) is clearly defined and finite at every non-transition point, as > > being, essentially, a finite sum at every such non-transition point. > > > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > I understand your logic, but the basis is incorrect. As WM > understandably complains, using the "completed set of naturals" as a > measure of anything simply does not work. It works in ZFC and NBG. That it does not work for TO or for "Mueckenh" (though for different reasons) is their problem, not ours. > You're focused on the Twilight > Zone. Even twilight beats the stygian darkness in which TO is operating.
From: Dik T. Winter on 8 Oct 2006 23:00 In article <1160310643.181133.6720(a)b28g2000cwb.googlegroups.com> mueckenh(a)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? > In German: Unendlichkeitsaxiom: Es gibt eine Menge, > die die leere Menge enth?lt, und wenn sie die Menge A enth?lt, so > enth?lt 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? > > You question whether "all x in N" does exist, apparently. Based on what? > > Based on the impossibility to index the positions of our 0.111..., False. > based on the vase, based on many other contradictions arising from "all > x in N do exist". False. No proof given. > > Yes, and that is false and not provable. > > A set containing all positions "up to position x" is a superset of a > set containing "position x". By what rule? What do you *mean* by "a set containing all positions..."? What do you *mean* by "a set containing...". I would state the the set {1, 2, 3, 4} contains all positions up to position 4, but that the set {4, 5} contains the position 4. But neither is a superset of the other. > A set containing "up to every position" defines a superset of set > containing "every position". But "every position" cannot be a proper > subset. Hence both sets are equivalent. Please first answer my question above, next, elaborate. > > I will re-iterate. Let L be your list (tacitly assumed to be unary > > representations of natural numbers), define a sequence of symbols such > > that it starts with 0., and for each n in N the n-th symbol is 1. I > > state that: > > (1) That sequence can be indexed, because the n-th symbol is indexed > > by the n-th item in your list. > > (2) For each n that sequence can be covered by the n-th item in your > > list upto the n-th symbol. > > (3) There is no n in your list that covers the whole of that sequence. > > Because if there were one such n, the (n+1)-st symbol is not covered. > > You claim (at various times) that either the definition is false, or that > > (1) is false. Well, clearly (1) is not false because of the definition. > > This definition "covers" only the naturals. If 0.111.. had only such > positions, ... Yes, it has. > Well, let us stop here. Well, you cn start stopping. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 8 Oct 2006 23:05 In article <4529afa4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45296779(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <452946ad(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > >>>>>> Less than any finite distance. Silly! > >>>>> And what is the smallest finite distance? > >>>>> > >>> Note question not answered!! But the correct answer of zero would have > >>> blown TO's argument to blazes, so one can see why he would not care to > >>> answer it. > >> I was away a couple days, but I answered this, not paying attention to > >> that apparent contradiction, since I don't consider 0 really a finite > >> number at all. It's a point with no measure, as every number is measured > >> relative to that point. > > > > If zero is not a number, how does TO keep the positive numbers separated > > from negatives? > > It's not a finite number. It's the origin. A finite number is a finite > distance from the origin. The origin is no distance from itself. Then TO's set of real numbers is two sets separated by a non-number? That does not match anyone else's set of reals. So TO casts himself again into outer darkness re res mathematical. > > >>>> When you claim that there are ordinals greater than any finite ordinal, > >>>> are you obligated to name the largest finite ordinal? > >>> When you claim there is a LUB to the reals strictly between 0 and 1, > >>> are you required to name the largest real strictly between 0 and 1? > >> No. That's my point. Why should I name the smallest object which is not > >> infinitesimal? > > > > That is not at all what I asked. So TO is doing his STRAW MAN fallacy > > thing again. > > You have no clue what the line of discussion was at this point, do you? I have no idea what TO is talking about, and am reasonably sure he doesn't either. > > >>> A "LUB" of the naturals does not have to be a natural any more than the > >>> LUB of the the reals strictly between 0 and 1 has to be a real strictly > >>> between 0 and 1. > >> If it's a discrete set, then I disagree. > > > > The set {(n-1)/n: n in N} is a discrete set with a LUB which is not a > > member of the set. In fact every strictly increasing sequence having a > > LUB has a LUB which is not a member of the sequence. > > That is not "the reals strictly between 0 and 1" but a subset thereof. So there is still no element within either set which is its LUB.
From: Virgil on 8 Oct 2006 23:13 In article <4529b325(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > If noon is not a point of transition, then nothing can change at noon, > and the vase is non-empty at every point before noon, so it must be > non-empty at noon as well. The function's value at noon can hardly be determined by values all of which are separated from noon by points of non-definition of the function, especially when the values on either side of every such point of non-definition are necessarily different.
From: David Marcus on 9 Oct 2006 01:13
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? -- David Marcus |