An uncountable countable set
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From: Mike Kelly on 5 Oct 2006 05:43 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <45215d2f(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Han de Bruijn wrote: > >>>>> stephen(a)nomail.com wrote: > >>>>>> how many balls are in the vase at noon? > >>>>>> > >>>>>> What does your "mathematics" say the answer to this > >>>>>> question is, in the "limit" as n approaches infinity? > >>>>> My mathematics says that it is an ill-posed question. And it doesn't > >>>>> give an answer to ill-posed questions. > >>>>> > >>>>> Han de Bruijn > >>>>> > >>>> Actually, that question is not ill-posed, and has a clear answer. The > >>>> vase will be empty, if there is any limit on the number of balls, and > >>>> balls can be removed before more balls are added, but it is not the > >>>> original problem, which states clearly that ten balls are inserted, > >>>> before each one that is removed. That's the salient property of the > >>>> gedanken. Any other scheme, such as labeling the balls and applying > >>>> transfinitology, violates this basic sequential property, and so is a ruse. > >>> One can pose any gedanken one likes. > >>> > >>> If TO does not like to be able to tell one ball from another, he does > >>> not have to play the game, but he should not ever try to pull that in > >>> games of pool or billiards. > >> If distinguishing balls gives a less exact answer, > > > > Less exact how? Ahem! No answer to a request for clarification!? > >> and a nonsensical one to boot > > > > It makes sense to me that if you put a ball into a vase and later > > remove it then it isn't there. It also makes sense to me that if you > > put a ball in a vase and don't remove it then it is still there. What > > *doesn't* make sense to me is that if you put some number of balls in a > > vase and remove them all then there are still some left. That seems to > > be what you are claiming. > > > > Note : I agree with those who say it makes no sense in physical terms > > to have an infinite number of balls. But mathematics is an idealisation > > so it can make sense to talk about the infinite, even if it is > > physically impossible. > > It makes no sense that adding ten balls and removing one will ever do > anything but increase the number of balls in the vase. Who ever said that at any "iteration" the number of balls in the vase doesn't increase? Strawman. >It makes no sense > to choose a unfounded theory Side note : Is this really coming from Tony Orlow? His theories are founded on "it feels nice to me". >over basic logic which states that if you > have 0 balls at any iteration, you had -9 balls in the previous. Whoever said you had 0 balls at an iteration? Strawman. You have 0 balls *at noon*. Noon is *not an iteration*. Do you seriously not understand this after having it explained so many times to you? Must be some sort of peculiar and persistant mental block. > It makes no sense to choose labels without end over infinite series. Then the original problem doesn't make sense. The original question includes the labels. It says at the nth iteration remove the nth ball. How do you remove the nth ball without knowing which ball the nth is? Spurious objection. > This theory is at odds with everything around it. It's at odds with your intuition. Excuse me while I fetch my violin. > > > > >> then that attention can be judged to be ill spent, and not > >> contributing to a solution at all. It is clear that sum(x=1->oo: 9) > >> diverges, is infinite, not 0. It's ridiculous to think otherwise. > > > > But the number of balls in the vase at noon *isn't* the limit of that > > sum, Tony. Nobody disagrees that that sum diverges (of course, we might > > disagree that it diverges to a "specific actually infinite value", but > > I digress...), people disagree that the limit of that sum is the same > > thing as the number of balls in the vase at noon. > > Add 10, remove 1, repeat. > (+10-1)+(+10-1)+.... > 9+9+.... How does that answer what I said in any way? That sum tells you nothing about what is in the vase at noon. Noon is not an iteration reached by that sum. > How many times were we doing this? Not enough for noon to be an iteration. >You can name the balls after Pokemon > for all I care, this sum doesn't not approach 0. I agree, that sum doesn't approach 0. But noon is not a part of that sum. Yes, at every iteration before noon the number of balls increases. No, there are not any balls in the vase at noon. Noon is distinct from the iterations. >Your labels are not math. The original question includes the labels. It says at the nth iteration remove the nth ball. How do you remove the nth ball without knowing which ball the nth is? Spurious objection. > > It seems a little barmy to spend a year arguing something that nobody > > disagrees with - that the sum 9, 18, 27... diverges. You should instead > > try to argue that the limit of that sum is equal to the number of balls > > in the vase at noon - that is what people are disagreeing with! Just > > saying "clearly" doesn't quite cut it. > > This thread was started by someone who agrees. Many agree. Han thinks the problem is nonsensical as stated. So does the thread starter, as far as I can tell. If one accepts that it makes sense to talk about the problem (even if it is physically unrealisable) then one accepts that you're adding an infinite number of balls before noon and removing each and every one before noon. What conclusion do you have other than that no ball is in the vase at noon? >Limits are math. Limit ordinals are not. Translation : I don't like them. > I'll think about the following, but have to get offline now. > > Later, > > Tony I look forward to it. -- mike.
From: Han.deBruijn on 5 Oct 2006 05:44 Mike Kelly schreef: > Pinpoint your beef with the following "method" : > > Problem : at one minute to noon, balls 1 thru 10 are added to the vase > and ball 1 is removed. At half a minute to noon balls 11 thru 20 are > added and ball 2 is removed. etc. > > Let noon = 0 and "one minute to noon" = -1. > > Let A(n,t) be 1 if the ball n is in the vase at time t, 0 if it is not > in the vase at time t. > > Let B(n) be the time that the nth ball is added to the vase and C(n) be > the time that it is removed. > > B(n) = -1/(2^(floor((n-1)/10))) > C(n) = -1/(2^(n-1)) > > Note that B(n) and C(n) are strictly less than 0. > > Now A(n,t) = { 1 if B(n) <= t < C(n) > 0 otherwise } Why? Let: Now A(n,t) = { 1 if B(n) <= t < C(n) undefined otherwise } > Note that A(n,0) = 0. Wrong. Note that A(n,0) = undefined. > Let S(t) be the number of balls in the vase at time t. Then > > S(t) = { sum(n=1..) A(n,t) } > > Then > > S(0) = { sum (n=1..) A(n,0) } > = { sum (n=1..) 0 } Wrong: = { sum (n=1..) undefined } > = 0 Wrong: = undefined > QED. QED. > > >>You can find this in any > > >>first year calculus text book. > > > > > > I have looked in several calculus books, starting with Apostol's, and > > > found no such thing in any of them. They are all careful to say that, > > > absent convergence, limit definitions say nothing about what happens. > > > > There are no other definitions of the infinite than limit definitions. > > There obviously *are* other definitions, you just do not approve of > them. Why should anyone particularly care for your likes and dislikes? Precisely! Why should anyone particularly care for your A(n,0) = 0 ? Han de Bruijn
From: Mike Kelly on 5 Oct 2006 05:46 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Quote [ Randy Poe ] : Physicists also > >> > >>>realize that things can exist in mathematics that aren't even > >>>approximations of a physical realizable. That aren't physically > >>>sensible in other words. > >> > >>That's only true for non-disciplinary mathematics. > > > > What is non-disciplinary mathematics? Is it not mathematics? > > It is mathematics without the discipline. You're being clear as mud but I'm going to take a wild guess and assume that means "mathematics that isn't about physically realisable things". So we have your claim "the only mathematics that is about non-physically realisable things is the mathematics that is about non-physically realisable things". Deep. Second guess was "mathematics that is not liked by Han". -- mike.
From: mueckenh on 5 Oct 2006 06:24 Dik T. Winter schrieb: > In article <1159649021.675137.307470(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > > So, again, no definition. Where did I not speak the truth? > > > > > > > > Here: "...because he never answered to questions about it". > > > > > > You gave on Usenet a definition of natural number in answer to my question > > > for it. I posted questions about that definition, and you never answered > > > them. So in what way is it a lie when I state that you never answered > > > them? > > > > I did not see an unanswered question. > > Apparently you did not see the questions at all. You stated as your definition > something like: > "numbers are in trichotomy with each other" > and I answered something like "so omega is a number". But I never did see an > answer. Omega is not in trichotomy with 1 + omega. > > I asked for a mathematical definition of number. You never gave one (and > also your paper does not give any mathematical definition), your only > mathematical definition was about the trichotomy. I am not sure about the definition of a number. Therefore I cannot give it. What I can give and have given, is a criterion what a number has to satisfy and I can give some examples which do not satisfy that criterion and which, therefore, are not numbers > > > Please repeat. > > If you do repeat. > > > By the way, please switch to the thread "Cantor > > Confusion" because this one has become too lengthy and, at home, I have > > only a slow internet access. So I am not able to follow this thread > > firmly. > > Oh. I did not know that slow internet access made long threads more > difficult to follow than short threads. It lasts a long while until all is loaded again, after I have posted. This problem, however, exists only during the weekends. But this thread is too long. I am not able and willing to read all contributions. > I see no reason to shift the > subject. And certainly not to a subject for which a thread already does > exist. The original subject of this thread is no longer under discussion here. Regards, WM
From: mueckenh on 5 Oct 2006 06:35
Dik T. Winter schrieb: > > > > (There are exactly twice so much > > > > natural numbers than even natural numbers.) > > > > > > By what definitions? You never state definitions. > > > > By the only meaningful and consistent definition: A n eps |N : > > |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > > Do you challenge its truth? > > No, I never did. But you draw conclusions about it about the set N. Indeed, > for each finite n, it is true. And N is nothing but the collection of all finite n. > > Your main argument and my main target is the full presence of all > > digits and their indexibility by numbers all of which do not cover > > 0.111... . > > You attack the existence of an infinite sequence, and so also the existence > of an infinite set. So be it. But that is just a negation of the axiom of > infinity. With that axioms such things do exist. The axiom of infinity does only state n+1 exists if n is given. It is realized by he numbers of my list. > > I mean: Indexing of all digit position of 0.111... by the unary numbers > > 0.1, 0.11, 0.111, ... is impossible unless all digi positions f > > 0.111... are also covered by these unary numbers. > > Again, confused. Each digit position can be covered, but there is no entry > in the list that covers all digit positions. On the other hand, every digit > can be indexed (when we interprete the elements of the list as unary natural > numbers). > > > Instead of "to index position" we can also say "to cover up to position > > n". Hence you assert that it is possible to cover 0.111... up to every > > position but it is impossible to cover every position. > > Yes. That is obviously a false claim, because "up to every (including this)" means "every". At this point a further discussion is really fruitless. > > > > > Take into account that also Cantor's diagonal argument cannot be > > > > executed in unary representation. > > > > > > Two red herrings in a single sentence. Can you get more? > > > (1) Cantor's diagonal argument was about countable sequences of two > > > symbols. There is only one countable sequence of one symbol. > > > > Cantor's argument was about reals. He strived for generality but did > > not see that two symbols are not enough. > > You are seriously wrong. Read his first paper. He treats numbers and rational functions. Nothing more. Regards, WM |