An uncountable countable set
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From: Tony Orlow on 9 Oct 2006 11:16 Dik T. Winter wrote: > 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? He means that the axioms can be stated without using universal quantifiers or existential quantifiers, as simple logical implications between propositions, I believe. WM, am I close? > > > 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? It can be stated as A eps S -> A U {A} eps S, with no quantifiers whatsoever and no loss of information. > > > > 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. What type of number can index the bit positions in the list of all binary-coded naturals? A finite number of bits is too few to produce a countably infinite set of strings, but a countably infinite set of bits produces an uncountably infinite set of strings. There is no way to remove bits or strings to get the countably infinite set, except to apply the restriction of finiteness on all strings, further destroying any particular measure of the set. > > > 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. I thought it was rather obvious that WM meant the set containing ONLY position x. So, take it that way and comment. > > > 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. Over and out. TOny
From: Tony Orlow on 9 Oct 2006 11:18 Virgil wrote: > 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. It's a 1-D continuum with an origin, a metric space. >>>>>> 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. Then don't make yourself look silly defending questions and comments that are irrelevant. >>>>> 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. If the Finlayson reals are used, indeed the LUB is the maximal member of the set of reals in [0,1). Ross, is that correct? Tony
From: Tony Orlow on 9 Oct 2006 11:21 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. TOny
From: Tony Orlow on 9 Oct 2006 11:25 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: > >> Sure, that sounds okay to me. In the naturals, the first is 1, the >> second 2, etc. What is the aleph_0th? > > I don't know. What is the "3/2"th natural number? > > Cheers - Chas > Is 3/2 even possibly the size of a set? The set of consecutive naturals from 1 through n has size n. The set of the first n naturals has maximal element n. Is N the set of the first aleph_0 naturals? If so, then aleph_0 must be the maximal element, but there is no maximal element. Therefore there is no size. Anything you can say about the maximal element is true of the size and vice versa. They are the same thing. Tony
From: Tony Orlow on 9 Oct 2006 11:27
Virgil wrote: > In article <e3ab0$4529ffd9$82a1e228$22026(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >> Virgil wrote: >> >>> In article <1160332251.241188.301420(a)i3g2000cwc.googlegroups.com>, >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>> But ah, a picture says more than a thousand words: >>>> >>>> http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg >>> But the function one should be graphing is not continuous at any of the >>> time of entry or exit of any ball, nor at noon (except it is >>> right-continuous at noon). >> Correct. I have "continuized" the function. The true (discrete) function >> should be like a staircase with a stair at each red ball. But a discrete >> version likewise explodes (i.e. not implodes) at noon. > > Actually not precisely "at" noon. And not before and not after. Like, you mean, never, right? > > 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. Thent he vase does not empty at noon, nor does it empty before, or after. Got an option 4? > > 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. > > 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 |