An uncountable countable set
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From: David Marcus on 28 Oct 2006 16:04 Randy Poe wrote: > Lester Zick wrote: > > On Fri, 27 Oct 2006 14:23:58 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > >Lester Zick wrote: > > >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > > >> >A very simple example is that there exists a smallest positive > > >> >non-zero integer, but there does not exist a smallest positive > > >> >non-zero real. > > >> > > >> So non zero integers are not real? > > > > > >That's a pretty impressive leap of illogic. > > > > "Smallest integer" versus "no smallest real"? Seems pretty clear cut. > > You must be joking. I can't believe even you can be this dense. You think Lester is trolling? > Is 1 the smallest positive non-zero integer? Yes. > > Is it the smallest positive non-zero real? No. 1/10 is smaller. > Ah well, then is 1/10 the smallest positive non-zero real? No, > 1/100 is smaller. Is that the smallest? No, 1/1000 is smaller. > > Does that second sequence have an end? Can I eventually > find a smallest positive non-zero real? > > How about the first? Is there something smaller than 1 which > is a positive non-zero integer? -- David Marcus
From: imaginatorium on 28 Oct 2006 16:07 Randy Poe wrote: > Lester Zick wrote: > > On Fri, 27 Oct 2006 14:23:58 -0400, David Marcus > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > > >Lester Zick wrote: > > >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > > >> >A very simple example is that there exists a smallest positive > > >> >non-zero integer, but there does not exist a smallest positive > > >> >non-zero real. > > >> > > >> So non zero integers are not real? > > > > > >That's a pretty impressive leap of illogic. > > > > "Smallest integer" versus "no smallest real"? Seems pretty clear cut. > > You must be joking. I can't believe even you can be this dense. Hmm, you must be new to Usenet? (Lester has just informed me - no idea why, since I didn't ask him - that he considers himself an original thinker.) > Is 1 the smallest positive non-zero integer? Yes. > > Is it the smallest positive non-zero real? No. 1/10 is smaller. > Ah well, then is 1/10 the smallest positive non-zero real? No, > 1/100 is smaller. Is that the smallest? No, 1/1000 is smaller. > > Does that second sequence have an end? Can I eventually > find a smallest positive non-zero real? > > How about the first? Is there something smaller than 1 which > is a positive non-zero integer? You also must be unaware that this is a poetry thread... Brian Chandler http://imaginatorium.org I wish I were a little grub With whiskers round my tummy I'd climb into a honeypot And make my tummy gummy
From: Virgil on 28 Oct 2006 16:14 In article <454364ae(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Apparently you are not aware of my > position on the subject. Bijections alone do not prove equinumerosity > for infinite sets. If they do not then nothing does. > Cardinality is a rough measure of equivalence class, > not a precise measure of the size of a set. There is no better measure. > In order to precisely > compare such infinite sets of values, one must measure over a common > infinite value range formulaically. Except that TO has never proved that his "formulaic measures" form a proper partial order relation on sets the way cardinality does. For cardinality, one can easily show that if |A| >= |B| then it is false that |B| > |A|. Can TO prove a similar result for his "formulaic measures"? At any rate he has never done so. And absent such a proof, and other proofs necessary for a partial ordering, his "formulaic measures" are, at best, dubious.
From: Virgil on 28 Oct 2006 16:21 In article <45436656(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > > <snip> > >>> Now correct me if I'm wrong, but I think you agreed that every > >>> "specific" ball has been removed before noon. And indeed the problem > >>> statement doesn't mention any "non-specific" balls, so it seems that > >>> the vase must be empty. However, you believe that in order to "reach > >>> noon" one must have iterations where "non specific" balls without > >>> natural numbers are inserted into the vase and thus, if the problem > >>> makes sense and "noon" is meaningful, the vase is non-empty at noon. Is > >>> this a fair summary of your position? > >>> > >>> If so, I'd like to make clear that I have no idea in the world why you > >>> hold such a notion. It seems utterly illogical to me and it baffles me > >>> why you hold to it so doggedly. So, I'd like to try and understand why > >>> you think that it is the case. If you can explain it cogently, maybe > >>> I'll be convinced that you make sense. And maybe if you can't explain, > >>> you'll admit that you might be wrong? > >>> > >>> Let's start simply so there is less room for mutual incomprehension. > >>> Let's imagine a new experiment. In this experiment, we have the same > >>> infinite vase and the same infinite set of balls with natural numbers > >>> on them. Let's call the time one minute to noon -1 and noon 0. Note > >>> that time is a real-valued variable that can have any real value. At > >>> time -1/n we insert ball n into the vase. > >>> > >>> My question : what do you think is in the vase at noon? > >>> > >> A countable infinity of balls. > > > > 1) It's not clear to me what you mean by that phrase but I'll assume > > the standard definition. Still, the question remains of which balls you > > think are in the vase? Does every natural number, n, have a ball in the > > vase labelled with that n? > > Conceptually, sure. > > > > > 2) How come noon "exists" in this experiment but it didn't exist in the > > original experiment? Or did you give up on claiming noon doesn't > > "exist"? What does that mean, anyway? > > Nothing is allowed to happen at noon in either experiment. They both end > up with countably many balls in the vase at noon. The experiment's > stated sequence logically precludes that the vase become empty. > > > > >> This is very simple. Everything that occurs is either an addition of ten > >> balls or a removal of 1, and occurs a finite amount of time before noon. > >> At the time of each event, balls remain. At noon, no balls are inserted > >> or removed. The vase can only become empty through the removal of balls, > >> so if no balls are removed, the vase cannot become empty at noon. It was > >> not empty before noon, therefore it is not empty at noon. Nothing can > >> happen at noon, since that would involve a ball n such that 1/n=0. > > > > Well, we're skipping ahead here much faster than I think will prove > > productive. Let's see.... > > > > By this logic, there is not a countable infinity of balls in the vase > > at noon in the new experiment I proposed. Everything that occurs is an > > addition of a single ball. At the time of each event, a finite number > > of balls are in the vase. At noon, no balls are inserted. If no balls > > are inserted at noon, the vase has the same state as before noon - a > > finite number of balls. > > > > Very good! There are an unboundedly large, but finite, number of finite > natural numbers. At every moment before noon, there is a boundedly large finite number of balls in the vase. If that situation is to carry over to noon, the number is still boundedly large at noon. Particularly since, outside of TO's twilight zone, there is no such thing as an unboundedly large finite number. So the options are boundedly large and finite or infinite. At least outside of TO's dreams.
From: Virgil on 28 Oct 2006 16:28
In article <45437ec0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > >> Ever heard of algebra or formulas? Ever seen a mapping between two sets > >> of numbers? > > > > This is a lame insult and irrelevant comment. It says nothing > > about what a "forumulaic relationship" between sets is. > > > > What is there to say? You know what a formula is. TO has as yet to show that his "formulaic relationships" provide a partial ordering on sets in the way that cardinalities do. In particular TO must prove that that if }A| >= |B| in his ordering then |B| > |A| is not possible, as well as all the other properties of a partial ordering. |