An uncountable countable set
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From: Tony Orlow on 4 Oct 2006 10:47 David R Tribble wrote: > Tony Orlow wrote: >>> On the other hand >>> I don't know why I said "neither can the reals". In any case, the only >>> way the ordinals manage to be "well ordered" is because they're defined >>> with predecessor discontinuities at the limit ordinals, including 0. >>> That doesn't seem "real" > > Virgil wrote: >>> In what sense of "real". There are subsets of the reals which are order >>> isomorphic to every countable ordinal, including those with limit >>> ordinals, so until one posits uncountable ordinals there are no problems. > > Tony Orlow wrote: >> In the sense that the real world is continuous, and you don't just have >> these beginnings with nothing before them. The real line is a line, with >> each point touching two others. > > That's a neat trick, considering that between any two points there is > always another point. An infinite number of points between any two, > in fact. So how do you choose two points in the real number line > that "touch"? > They have to be infinitely close, so actually, they have an infinitesimal segment between them. :)
From: Tony Orlow on 4 Oct 2006 10:54 Mike Kelly wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> Han de Bruijn schrieb: >> >>> stephen(a)nomail.com wrote: >>> >>>> Han.deBruijn(a)dto.tudelft.nl wrote: >>>> >>>>> Worse. I have fundamentally changed the mathematics. Such that it shall >>>>> no longer claim to have the "right" answer to an ill posed question. >>>> Changed the mathematics? What does that mean? >>>> >>>> The mathematics used in the balls and vase problem >>>> is trivial. Each ball is put into the vase at a specific >>>> time before noon, and each ball is removed from the vase at >>>> a specific time before noon. Pick any arbitrary ball, >>>> and we know exactly when it was added, and exactly when it >>>> was removed, and every ball is removed. >>>> >>>> Consider this rephrasing of the question: >>>> >>>> you have a set of n balls labelled 0...n-1. >>>> >>>> ball #m is added to the vase at time 1/2^(m/10) minutes >>>> before noon. >>>> >>>> ball #m is removed from the vase at time 1/2^m minutes >>>> before noon. >>>> >>>> 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. >> You are right, but the illness does not begin with the vase, it beginns >> already with the assumption that meaningful results could be obtained >> under the premise that infinie sets like |N did actually exist. > > The meaningful result is that if you allow "|N exists" then the vase > empties at noon. Even if you don't allow that in your mathematics, you > can surely accept the logical conclusion that IF you allow that THEN > the vase is empty at noon. No? Only if you change the order of events, or refuse to say when the vase empties or how. Any "|N" aside, the problem clearly states that ten balls are added and then one removed, per iteration, so if the vase emptied, it could only be with the removal of that 1 ball, not with the addition of the ten balls, since that would require that there had been -10 balls in the vase. But, for there to be 1 ball left, which when removed left an empty vase, ten would have been inserted right beforehand, meaning there had to have been -9 balls in the vase. Neither negative count is possible, therefore the vase could not have emptied. When you come to two logical conclusions given two lines of thought, how do you resolve that? > > There is no logical contradiction in the existence of |N, by the way. > You're free to reject it in your versions of mathematics (which I have > no doubt are of great use and value) but it's absurd to claim that > mathematics that does accept it is meaningless. > > If you are arguing that the existence of |N is contradictory and > anything derived from it is meaningless then what is the point of > arguing about the vase problem? What is the point of arguing about the > random natural problem? You don't accept a premise that those problems > mean nothing without. Restrict yourself to attempting to show that the > axiom of infinity (or existence of |N, if you like) is contradictory. >
From: Tony Orlow on 4 Oct 2006 11:03 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? > >> 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. It makes no sense to choose a unfounded theory over basic logic which states that if you have 0 balls at any iteration, you had -9 balls in the previous. It makes no sense to choose labels without end over infinite series. This theory is at odds with everything around it. > >> 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 many times were we doing this? You can name the balls after Pokemon for all I care, this sum doesn't not approach 0. Your labels are not math. > > 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. Limits are math. Limit ordinals are not. I'll think about the following, but have to get offline now. Later, Tony > > Here's my take on things... > > 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 } > > Note that A(n,0) = 0. > > 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 } > = 0 > > QED. > > Now I'm going to look a little at your argument, based on limits. > > Let's look at the sequence of S_ns where > > S_n = S(-1/(2^(n-1)) > > That is, S_n is the number of balls in the vase "just after" the nth > iteration. Then, you say, the limit of the sequence of S_ns must equal > S(0) (ie, the number of balls at noon) and the limit of the sequences > of S_ns (9, 18, 27,...) is "positive infinity" and thus the number of > balls at noon is infinite. But this is fallacious reasoning, Tony! What > justification do you have for asserting that the limit of the S_ns is > equal to S(0)? Presumably something along the lines of... > > + oo = lim((n->oo), S_n) > = lim((n->oo), S(-1/(2^(n-1))) > = S(lim(n->oo), -1/(2^(n-1))) > = S(0) > > But this is not a valid manipulation of limits. lim(n->oo,S(T(n))) = > S(lim(n->oo),T(n)) is true when S is continuous. But S is not > everywhere continuous in our problem. So you have no justification for > your assertion that S(0) is equal to the limit of the S_ns. >
From: MoeBlee on 4 Oct 2006 11:05 Tony Orlow wrote: > That's because the current view of the naturals is as containing only > finite values. Not because it's a "view" but because it is a theorem. A set that has zero as a member, and such that every member is a successor except for zero, and that satisfies the induction schema is a set that has only finite members. A theorem, not a "view". Go figure. > > , but if there are an infinite number of > >> naturals generating using increment, then there are naturals which are > >> the result of infinite increments, > > > > It does not follow that having an endless supply of something means that > > any of them are infinite. > > A contradiction follows from having an infinite number of elements each > separated from its closest neighbors by a unit of difference, without > there being two elements with an infinite difference between them. A CONTRADICTION? From what AXIOIMS? Please show a derivation of a sentence P and ~P from the axioms. Oh, that's right, by "contradiction" you don't mean a contradiction in the sense of a sentence and its negation; you mean something that doesn't sit with your personal intuition. > If > every pair of naturals are within a finite difference of each other, > Then there is no infinite count in either direction, quantitatively. The > reason it fits the Dedekind definition is that there is no largest > finite, but with the identity relation between element count and value > in the reals, the set cannot be said to be quantitatively infinite, but > merely unbounded and denumerable. Countable infinity is potential, not > actual. Right, in other words, you want to define certain words differently. > That ignores sum(n=1->oo: 1)=oo, You ignore that the above notation is meaningless. > and it rests on the notion that all > naturals are finite, when proving its own premise. No, in set theory, other than AXIOMS, no formula is used in a proof of itself. > No, "is finite" is not an equality. That's true. But what of it? > Not when applying induction to the infinite case. How about applying infduction to the inducfinite case. Have you looked into that? There's fellow named Ony Torlow who has these wonderful theories about it. He won't tell anyone what his axioms, primitives or definitions are, but he claims it's a wonderful theory nonetheless, and, of course, we all follow his theory because he's just so very positive sounding when he says things like "not when applying infduction to the indcufinite case". > You can prove > equalities or inequalities. Equalities between expressions hold true in > the infinite case. Where inequalities are concerned, one must make sure > that the difference which establishes the inequality does not have a > limit of 0 as n->oo. Otherwise, your inequality holds only for finite n. > In this case, the inequality is "n<oo", where "oo" means "any infinite > number". Clearly lim(n->oo: n) is not less than oo. Right, and the derivative of the quantitative equiquality is clearly floor(k < oo, for oo -> ky). > So, limits are nonsense? Your use of 'lim' sure is. > I don't need to escape it. It's purring in my lap. Why don't we keep whatever is going on in your lap out of this, okay? > I have set forth good > rules for applying inductive proof to these numbers. The only decent > attempt at refutation was Chas' staircase, but that also boils down to a > matter of the definition of the limit, in a measureless point set > topology. That refutation was refuted, and the only other attempt used a > nested limit of 0 difference to produce a discontinuity at x=0, which > I've said makes the proof valid only for the finite case. Bravo! Well done, Tony! MoeBlee
From: Ross A. Finlayson on 4 Oct 2006 11:28
Han de Bruijn wrote: > imaginatorium(a)despammed.com wrote: > > > Han de Bruijn wrote: > > > >>The question is: how many balls are there in the vase at noon. > >>This question is meaningless, because noon is never reached. > > > > Really? When's lunch, then? > > Time is _suggested_, but not present, in the Balls in a Vase problem. > > Han de Bruijn It certainly is, time. The function that is the count of balls in the vase is a function of time. Use is made of points at infinity in the (projectively extended) natural and real numbers quite often. Some transformists use it to divide by zero, in a specialized way dependent on systemic conditions of the numerator. Sometimes transfinite (infinite) induction works, other times it doesn't, the transfer principle doesn't always hold, for each of various systemic elements. The balls and vase is cast as a thought experiment, similar to the three people who pass a gold bar in a circle, where whichever philosopher holds the bar at time 1-1/2^n hands it to the next, and the question is: at noon, who holds the bar? (Each, none, uniform random.) As a thought experiment, it's possible to offput some arguments along the lines of that there isn't a completed infinity, but in so doing then that seems to allow the user to set other realistic-seeming conditions, placing the thought experiment in a richer context, because the intuitive object is used to deduce characteristics in what would otherwise be an unreal situation. So, there is an indefinite supply of balls each one serially labelled which you can access in order. Put ten balls into the vase. Then, retrieve ball one. Do you just reach into the vase and pluck ball one? Is there a golem that catches ball 1, ball 2, and so on as they enter, either synchronized to the external world of the vase or, where there would eventually have to be infinitely golems as for however many finitely steps you pass there will be many more and each time a golem retrieves a ball the next ball's serial it would hold to later retrieve, the difference in serials, would increase by an ever-larger amount on each chance the golem has to be the retrieval agent? Can you look inside the base and grab the ball to be retrieved? How about this: the golem (demon, little automaton that follows your instructions as contrasts a fickle cat) has a little pen and a memory and every time you toss in ten, it relabels one of them to what you expect back and returns it, keeping the nine each time? Consider finite cases. Put in ten balls and get one back. Vase is non-empty. Basically I figure with some simple rules about the balls and vase, thought experiments about them can be more readily observed to match expectations of the mathematical objects. To retrieve ball 1, you have to pour all the balls out and inspect each one's label, or select one (or possibly several) at (uniform) random with/without replacement because you can't see into the vase or otherwise read the labels without withdrawing the labelled ball from the vase. I wonder, if the vase is opaque, or its opacity is a function of the data rate, would you be able to shake it to see if it's empty? What about if you can only empty the vase at the end, the vase is too heavy for you to lift. It may be a Rube Goldberg machine, just avoid the "here a miracle occurs" of the cartoon proof, where the guy says "you should expand this part here." If there are no balls in the vase at noon, you ran out of balls to put into the vase before that. When what you thought was the last ball was plucked, you just added ten since removing any. For those to occur at once, removing without adding, you lost distinction and ordering of events, in time: Zeno, HUP, blah blah bah, time. Ross |