An uncountable countable set
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From: cbrown on 26 Oct 2006 15:27 Tony Orlow wrote: > Mike Kelly wrote: <snip> > > My question : what do you think is in the vase at noon? > > > > A countable infinity of balls. > > 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. No one disagrees with the above statements. > The vase can only become empty through the removal of balls, Note that this is not identical to saying "the vase can only become empty /at time t/, if there are balls removed /at time t/"; which is what it seems you actually mean. This doesn't follow from (1)..(8), which lack any explicit mention of what "becomes empty" means. However, we can easily make it an assumption: (T1) If, for some time t1 < t0, it is the case that the number of balls in the vase at any time t with t1 <= t < t0 is different than the number of balls at time t0, then balls are removed at time t0, or balls are added at time t0. > 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. Now your logical argument is complete, assuming we also accept (1)..(8): If the number of balls at time t = 0, then by (7), (5) and (6), the number of balls changes at time 0; and therefore by (T1), balls are either placed or removed at time 0, implying by (5) and (6) that there is a natural number n such that -1/n = 0; which is absurd. Therefore, by reductio ad absurdum, the number of balls at time 0 cannot be 0. However, it does not follow that the number of balls in the vase is therefore any other natural number n, or even infinite, at time 0; because that would /equally/ require that the number of balls changes at time 0, and that in turn requires by (T1) that balls are either added or removed at time 0; and again by (5) or (6) this implies that there is a natural number n with -1/n = 0; which is absurd. So again, we get that any statement of the form "the number of balls at time 0 is (anything") must be false by reductio absurdum. So if we include (T1) as an assumption as well as (1)..(8), it follows logically that the number of balls in the vase at time 0 is not well-defined. Of course, we also find that by (1)..(8) and (T1), it /still/ follows logically that the number of balls in the vase at time t is 0; and this is a problem: we can prove two different and incompatible statements from the same set of assumptions So at least one of the assumptions (1)..(8) and (T1) must be discarded if we are to resolve this. What do you suggest? Which of (1)..(8) do you want discard to maintain (T1)? Cheers - Chas
From: Lester Zick on 26 Oct 2006 15:29 On 25 Oct 2006 17:47:15 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> Yeah, Moe, look you do a lot of talking to justify one primitive. > >I'm not justifying a primitive. Well I'm glad you admit it. I was too polite to suggest it myself. >> My >> primitive is "contradiction" or "differences" and I can justfy that as >> true in one sentence. You say all your contentions "revert" (whatever >> that may mean) to one "undefined non logical primitive" by which I >> assume you mean one outright non demonstrable assumption > >No, I don't mean that. And I've already explained the difference. You have? Well thanks for the advisory anyway. > And >again, since you are unfamiliar with even the most basic notions such >as a primitive symbol, I'm pretty much wasting my time every time I >write up explanations for you to receive in the lap of your own >ignorance. Look who's talking about mis statements and mis characterization now, Moe, not to mention outright character assassination.Cut to the quick. >> >No, primitives are individual symbols. >> >> So they're just symbolic assumptions of truth. > >No, they are very much NOT that. Then what are they, Moe? You're mighty long on flatout assertions when it comes to what I say but mighty short on justifications when it comes to what you say. >> >Yes, the term 'cardinality' is not in the above definition. >> >> So now if you don't say "cardinality" the concept isn't there? > >It just doesn't even work in the way you pose the question. I just >can't explain how to do certain tasks on a computer to someone who >won't even push the "on" button on the machine. I just can't explain >how to regard mathematical defintions using mathematical logic to >someone who just won't even read page one of a textbook. So now I can just add all this to the list of things you can't do but have no problem criticizing in others when they criticize you for things you can't do, can't justify, and can't explain. >> In other words if I can't intuit your faith based belief in standard >> set analytical techniques you need to get back to the practice of your >> religion and leave me to my own devices? > >No, my point is that I can't explain page 100 to you if you won't read >page 1. And I can't explain truth to you if you insist that it's preferable to assume the truth of your assumptions instead of demonstrating it. >Nor can I keep very interested in a conversation with someone such as >you asks me questions on the subject but then replies to my answers >with essentially, "Who asked you? Who cares?" Just trying to circumscribe the domain of ignorance with pointed questions, Moe, when it comes to your complaining about things others are ignorant of. Do you really expect anyone to research and document for you the claims you make for modern math when you admit right up front you can't justify their truth yourself? I mean if you can't even be bothered to justify your opinions of a subject you claim to be conversant with except by regression to other opinions about the same subject, why would you expect others to take your claims seriously? ~v~~
From: stephen on 26 Oct 2006 15:50 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> stephen(a)nomail.com wrote: >>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>>>> imaginatorium(a)despammed.com wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: >>>>>>>>> Tony Orlow wrote: >>>>>>>>>> As each ball n is removed, how many remain? >>>>>>>>> 9n. >>>>>>>>> >>>>>>>>>> Can any be removed and leave an empty vase? >>>>>>>>> Not sure what you are asking. >>>>>>>> If, for all n e N, n>0, the number of balls remaining after n's removal >>>>>>>> is 9n, does there exist any n e N which, after its removal, leaves 0? >>>>>>> I don't know what you mean by "after its removal"? >>>>>> Oh, I think this is clear, actually. Tony means: is there a ball (call >>>>>> it ball P) such that after the removal of ball P, zero balls remain. >>>>>> >>>>>> The answer is "No", obviously. If there were, it would be a >>>>>> contradiction (following the stated rules of the experiment for the >>>>>> moment) with the fact that ball P must have a pofnat p written on it, >>>>>> and the pofnat 10p (or similar) must be inserted at the moment ball P >>>>>> is removed. >>>>> I agree. If Tony means is there a ball P, removed at time t_P, such that >>>>> the number of balls at time t_P is zero, then the answer is no. After >>>>> all, I just agreed that the number of balls at the time when ball n is >>>>> removed is 9n, and this is not zero for any n. >>>>>> Now to you and me, this is all obvious, and no "problem" whatsoever, >>>>>> because if ball P existed it would have to be the "last natural >>>>>> number", and there is no last natural number. >>>>>> >>>>>> Tony has a strange problem with this, causing him to write mangled >>>>>> versions of Om mani padme hum, and protest that this is a "Greatest >>>>>> natural objection". For some reason he seems to accept that there is no >>>>>> greatest natural number, yet feels that appealing to this fact in an >>>>>> argument is somehow unfair. >>>>> The vase problem violates Tony's mental picture of a vase filling with >>>>> water. If we are steadily adding more water than is draining out, how >>>>> can all the water go poof at noon? Mental pictures are very useful, but >>>>> sometimes you have to modify your mental picture to match the >>>>> mathematics. Of course, when doing physics, we modify our mathematics to >>>>> match the experiment, but the vase problem originates in mathematics >>>>> land, so you should modify your mental picture to match the mathematics. >>>> As someone else has pointed out, the "balls" and "vase" >>>> are just an attempt to make this sound like a physical problem, >>>> which it clearly is not, because you cannot physically move >>>> an infinite number of balls in a finite time. It is just >>>> a distraction. As you say, the problem originates in mathematics. >>>> Any attempt to impose physical constraints on inherently unphysical >>>> problem is just silly. >>>> >>>> The problem could have been worded as follows: >>>> >>>> Let IN = { n | -1/(2^floor(n/10) < 0 } >>>> Let OUT = { n | -1/(2^n) } >>>> >>>> What is | IN - OUT | ? >>>> >>>> But that would not cause any fuss at all. >>>> >>>> Stephen >>>> >> >>> It would still be inductively provable in my system that IN=OUT*10. >> >> So you actually think that there exists an integer n such that >> -1/(2^floor(n/10)) < 0 >> but >> -1/(2^n) >= 0 >> ? >> >> What might that integer be? >> >> Stephen >> > How do you glean that from what I said? Your "largest finite" arguments > are very boring. How do I glean that? You claim that IN does not equal OUT. IN contains all n such that -1/(2^floor(n/10)) < 0 and OUT contains all n such that -1/(2^n) < 0 Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), so presumably IN is bigger than OUT, and IN contains elements that are not in OUT. The only way n can be an element of IN, but not of out is if -1/(2^floor(n/10)) < 0 but -1/(2^n) >= 0 So apparently you do not think such an n exists, yet you think there are elements in IN that are not in OUT. Those are contradictory positions. Stephen
From: Virgil on 26 Oct 2006 16:36 In article <4540cf58(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > imaginatorium(a)despammed.com wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> As each ball n is removed, how many remain? > >>>>> 9n. > >>>>> > >>>>>> Can any be removed and leave an empty vase? > >>>>> Not sure what you are asking. > >>>> If, for all n e N, n>0, the number of balls remaining after n's removal > >>>> is 9n, does there exist any n e N which, after its removal, leaves 0? > >>> I don't know what you mean by "after its removal"? > >> Oh, I think this is clear, actually. Tony means: is there a ball (call > >> it ball P) such that after the removal of ball P, zero balls remain. > >> > >> The answer is "No", obviously. If there were, it would be a > >> contradiction (following the stated rules of the experiment for the > >> moment) with the fact that ball P must have a pofnat p written on it, > >> and the pofnat 10p (or similar) must be inserted at the moment ball P > >> is removed. > > > > I agree. If Tony means is there a ball P, removed at time t_P, such that > > the number of balls at time t_P is zero, then the answer is no. After > > all, I just agreed that the number of balls at the time when ball n is > > removed is 9n, and this is not zero for any n. > > > >> Now to you and me, this is all obvious, and no "problem" whatsoever, > >> because if ball P existed it would have to be the "last natural > >> number", and there is no last natural number. > >> > >> Tony has a strange problem with this, causing him to write mangled > >> versions of Om mani padme hum, and protest that this is a "Greatest > >> natural objection". For some reason he seems to accept that there is no > >> greatest natural number, yet feels that appealing to this fact in an > >> argument is somehow unfair. > > > > The vase problem violates Tony's mental picture of a vase filling with > > water. If we are steadily adding more water than is draining out, how > > can all the water go poof at noon? Mental pictures are very useful, but > > sometimes you have to modify your mental picture to match the > > mathematics. Of course, when doing physics, we modify our mathematics to > > match the experiment, but the vase problem originates in mathematics > > land, so you should modify your mental picture to match the mathematics. > > > > I disagree. When you formulate a theory, whether scientific or > mathematical, the goal should be to draw conclusions in line with > observations. In science, it's no problem to disprove a theory, if there > is a verifiable situation which it predicts incorrectly. When it comes > to math, there is no such test, but the whole of mathematics should be > consistent, and where one theory contradicts another, that's an > indication that one or the other is less than correct. That depends. If the apparently contradictory results follow from different axiom systems, they may both be quite valid. > In the case of > questions regarding oo, no theory should cause blatant contradictions, > such as an event occurring but there being no moment in time during > which it is occurring. If you have to accept such a conclusion to > salvage a theory, it's time to look for alternatives that don't require > you to sacrifice common sense and basic logic. This is just one example > of where this theory goes wrong, along with proper subsets of the same > size as the superset, and the concept of a smallest infinity. I simply > don't accept the theory, because its conclusions are bizarre. And we do not accept TO's theories for quite the same reason, his conclusions are too bizarre. And many of his assumptions are too bizarre as well.
From: Virgil on 26 Oct 2006 16:38
In article <4540d016(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > It would still be inductively provable in my system that IN=OUT*10. That only shows that N = N. e\Ergo N\N = {}. |