An uncountable countable set
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From: imaginatorium on 17 Oct 2006 15:02 Lester Zick wrote: > On 17 Oct 2006 10:37:08 -0700, imaginatorium(a)despammed.com wrote: > > >Tony Orlow wrote: > > [. . .] > > >> Oh. What was Aleph_0 again? > > > >You have, I'm sure been told dozens, if not hundreds of times - Aleph_0 > >is the name for the cardinality one might explain to children as "you > >can count, and reach any of them, but the counting never stops". > > So now aleph is an empirical concept, Brian? No, Lester. Technically (I expect you'll enjoy hearing that word!), aleph is a letter in the Hebrew alphabet. At least I believe so - I do remember John Conway using beth in a lecture, and commenting that it's a pity people don't know more than one letter of Hebrew. > ... I mean unless you > consider children too lazy or stupid to intuit the subtle virtue of > set theoretic assumptions underlying their ability to count one by > one. Sentence is a bit long for me to grasp in one go, but no, I don't consider children too lazy or stupid for anything, in general. That generally happens when we get older. Brian Chandler http://imaginatorium.org
From: Lester Zick on 17 Oct 2006 17:50 On 17 Oct 2006 12:02:36 -0700, imaginatorium(a)despammed.com wrote: > >Lester Zick wrote: >> On 17 Oct 2006 10:37:08 -0700, imaginatorium(a)despammed.com wrote: >> >> >Tony Orlow wrote: >> >> [. . .] >> >> >> Oh. What was Aleph_0 again? >> > >> >You have, I'm sure been told dozens, if not hundreds of times - Aleph_0 >> >is the name for the cardinality one might explain to children as "you >> >can count, and reach any of them, but the counting never stops". >> >> So now aleph is an empirical concept, Brian? > >No, Lester. Technically (I expect you'll enjoy hearing that word!), Well TECHNICALLY, Brian, any concept you resort to empirical validation for is an empirical concept. Children learning to count is scarcely a mathematical validation. Just calling it a cardinality is no justification whatsoever since I rather doubt children know what cardinality means. >aleph is a letter in the Hebrew alphabet. At least I believe so - I do >remember John Conway using beth in a lecture, and commenting that it's >a pity people don't know more than one letter of Hebrew. I suspect you mean it's a pity modern mathematikers don't. >> ... I mean unless you >> consider children too lazy or stupid to intuit the subtle virtue of >> set theoretic assumptions underlying their ability to count one by >> one. > >Sentence is a bit long for me to grasp in one go, but no, I don't >consider children too lazy or stupid for anything, in general. That >generally happens when we get older. And for others it begins when children learn that aleph is their raison d'etre for learning to count. ~v~~
From: Virgil on 17 Oct 2006 17:56 In article <26453$4534c7d5$82a1e228$20375(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <45341a3a$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>Well, I think that, while the empty set may easily be taken to represent > >>0, 1 is not the set containing 0. That doesn't seem, even at first > >>glance, like a very accurate model of what 1 is. > > > > If TO is not happy with the set representing 1 containing a single item > > does TO want the set representing 1 to contain more or less that single > > item? > > That single item is the EMPTY set, pasted between curly braces. HdB is missing my point here. If TO accepts {} as representing 0 but does not like {{}} as representing 1, what does TO suggest replace {{}} as representing1?
From: cbrown on 17 Oct 2006 18:19 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> Tony Orlow wrote: > >>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> cbrown(a)cbrownsystems.com wrote: > >>> <snip> <snipitty-snip> > > Do you accept the above statements, or do you still claim that there is > > /no/ valid proof that ball 15 is not in the vase at t=0? > > > > 15 is a specific finite number for which we can state its times of entry > and exit. Agreed, > At its time of exit, balls 16 through 150 reside in the vase. Agreed. > For every finite n in N, upon its removal, 9n balls remain. "upon its removal" = "at the time of ball n's removal"; Agreed. > For every n > e N, there is a finite nonzero number of balls in the vase. "For every n e N, there is a finite non-zero number of balls in the vase at t = -1/n". Agreed. > Every > iteration in the sequence is indexed with an n in N. "Balls are only added or removed at a time t = -1/n for some natual n." Agreed. > Therefore, nowhere > in the sequence... ...., i.e, at no time t such that t = -1/n for some natural n, ... > is there anything other than a finite nonzero number of > balls in the vase. Agreed. > > Now, where, specifically, in the fallacy in that argument? > Well, what do you state is the conclusion of this argument? If the conclusion of this argument is "we cannot therefore state that ball 15 is not in the vase at t=0", I really don't see how you have addressed the issue. You agree that ball 15 is removed, and not put back in the vase at any time before or at noon; and I think you would agree that if if a ball is not put in the vase, it cannot be in the vase. Therefore ball 15 is not in the vase at noon; and nothing you said above challenges the logic of this conclusion. If your conclusion is "therefore, at t=0, there must be a finite nonzero number of balls in the vase", then the fallacy is called non sequituur - it doesn't logically follow. * Because t=0 is /not/ a time such that t = -1/n for some natural n; * Therefore your statements regarding exclusively times t that /are/ of the form -1/n for some natural n do not /automatically/ apply to t=0. In order to make a conclusion about t=0 from your statements, you must appeal to some /other principle/ which connects the state of the vase at times t = -1/n for natural number n; with the state of the vase at times which are /not/ of the form t = -1/n for some natural number n; such as t=-2/3 or (more saliently) t=0. > >> Your statement concerning n does > >> not cover noon, because noon=f(oo), and oo is outside your range. > > > > You've lost me. > > Nothing happens at noon, if all sequential iterations are finite, given > the time sequence. That is not inconsistent with the statement "ball 15 is not in the vase at noon." > At all moments before noon, as has been conceded, > there are a nonzero number of balls in the vase. > That is not inconsistent with the statement "ball 15 is not in the vase at noon." > > > > What is f? What does it mean to say "noon = f(oo)"? How does this > > disprove the assertion that ball 15 is not in the vase at t=0? > > > > It means that every finite iteration occurs before noon, so the only > ones that can happen AT noon are infinite. The only "iterations" that occur are associated with natural numbers; there are no "infinite iterations" at all that "happen" at any time. > You have no infinite > iterations... .... just as I have no solid gold statuettes of Richard Nixon kissing Henry Kissinger... > ... so noon does not occur, What on earth does that even mean? Where, in the problem statement, do we conclude that "t=0" is not something that can "occur"? Which values of t can "occur"? The fact that 3/2 is not a natural number makes the statement "when n=3/2" nonsensical; but it doesn't follow that the statement "when t= -1/(3/2) = -2/3" is therefore also nonsensical. Equally, the fact that oo is not a natural number makes the statement "when n=oo" nonsensical; but it doesn't follow that the statement "when t=0" is therefore also nonsensical. > ... at at every moment BEFORE noon there > is a nonzero number of balls in the vase. Agreed. So what is required is a mathematical argument that is of the form: (i) We agree that A, B, C, .. and so on. (ii) We demonstrate that that (A and B and C and so on) implies X. (iii) Therefore, we conclude that X is true. What is currently is /not/ lacking is statements such as A, B, C which we agree on; e.g., "when ball 15 is removed, there are 135 balls in the vase". What /is/ lacking is a valid argument that states, for example: /because/ there are 135 balls in the vase when ball 15 is removed, /therefore/ ball 15 is in the vase at time t=0. > >> So, > >> you really don't have any claim with regard to what happens at noon. Its > >> beyond your purview. > > > > On the contrary, in a mathematical sense, a thing "happens" (i.e., can > > be concluded from the problem statement) if, and /onl/ if, it can be > > logically deduced from assertions in the problem. > > Deduction depends on assumptions. Correct. > Set theory's are phony in this case. Set theory is not making assumptions here; /we/ are making assumptions here. Which assumptions are reasonable? I make the following assumptions: (1) When we speak of a time t, we mean some real number t. (2) If a ball is in the vase at any time t0, there is a time t <= t0 for which we can say "that ball was placed in the vase at time t". (3) If a ball is removed from the vase at time t1, and there is no time t such that t1 < t <= t2 when that ball is placed in the vase, then that ball is not in the vase at time t2. (4) If a ball is placed in the vase at time t, it must be in accordance with the description given in the problem: it must be a ball with a natural number n on it, and the time at which it is placed in the vase must be -1/floor(n/10). (5) If a ball is removed from the vase at time t, it must be in accordance with the description given in the problem: it must be a ball with a natural number n on it, and the time at which it is removed from the vase must be -1/n. (6) If n is a natural number, then the ball labelle
From: David Marcus on 17 Oct 2006 20:49
Ross A. Finlayson wrote: > David Marcus wrote: > > MoeBlee wrote: > > > David Marcus wrote: > > > > Ross A. Finlayson wrote: > > > > > There is no universe in ZF, ZF is inconsistent. > > > > > > > > What exactly is the inconsistency, please? > > > > > > Of course he'll never show you an inconsistency. "Set theory is > > > inconsistent" is just one among the phrases that he's fond of > > > muttering. > > > > Apparently true. > > > > -- > > David Marcus > > No, I think I already have, but you won't accept it, because you're > invincible, in parrradise. Also, I do not "mutter". > > Quantify over sets: where do they come from? If you think it's the > cumulative hierarchy, the axiom of infinity says it's all of them. > > Quantify over sets, it's not a set. There's no universe in ZF, and > there is a universe. So, ZF's implication that it describes the > universe of sets is obviously wrong. > > Also, I argue that incompleteness is inconsistency. > > In pure set theory, everything's a set. You may not mutter, but I can't make any sense of this. Please answer this question: are you saying that ZF is inconsistent (where the word "inconsistent" has its standard mathematical meaning, i.e., proves both P and not P for some P) or are you saying something else? A yes or no answer would be good. -- David Marcus |