Wilhelm

@Paul: It is clearly not physical. Therefore you should ask. But that is irrelevant since it is not mathematical either. When by definition every step n is followed by another step n+1, then it is impossible to get rid of the finity. And for finity we have an increase. The "final" state empty urn violates the basic definition. Your argument of 1/2 + 1/4 + 1/8 + ... is inappropriate. Movement does not work as Zeno thought. Easiest proof: Try it starting at the other side.

@Paul: I would recommend that you start an inquiry among physicists (and even mathematicians) of your acquaintance whether they endorses RL as being realistic.

@Paul: In physics and mathematics we can use infinite models (space, wires) but only if the function (force of gravity or electric force) exerted on the sample vanishes with increasing distance. Same in mathematics. Functions like 1/r^2 have a limit. Your example however applies infinitely many balls in an row, i.e., in an infinite distance (because each one has a finite diameter in containing a clockwork) where all balls exert the same action when their time has come.

@Carl: Of course it is impossible to define a real number by an infinite string of digits without having a finite formula. Therefore set theory is easily to be contradicted. There is the same naivety as in believing that the removal of every ball leaves the empty urn. Every ball is followed by infinitely many. That is infinity!

The RL-procedure is therefore completely theoretical. And the main point of this theory is the definition that only at finite steps transfer of balls is possible. If the limit is claimed to be empty by set theory but all finite steps are not empty, then the limit is a different state that somehow must come into being. By definition this is excluded, during the possible steps. Therefore the set-theoretic claim is contradicted.

Infinite numbers of balls are absolutely unphysical because the accessible universe has a finite mass and a finite number of separable particles. Further the highest velocity is c and the smallest distance is 10^-95 m, the wavelength of a photon containing the whole mass of the accessible universe. Further the smallest time interval is Planck time 10^-44 s.

Rigorous mathematics is fine, but when a theory has been contradicted then it has to be dropped, Set theory is contardicted by the limit { } of the sequence {1}, {2}, {3}, ... counting the revolutions of an eternal merry-go-round. The empty limit is not only violating the logic of induction (every rotation is followed by another one) but even set theory itself, since when we count the revolutions {1}, {1}, {1}, ... the the limit is{1] and not empty.

"History shows that when mathematics demonstrates something rigorously, it's best to take it seriously". An example is the transfer of natural numbers 1, 2, 3, ... from urn A to urn C via an intermediate urn B in such a way that n must not leave B before n+1 has entered B. Then by strict mathematical definition B will never be empty and C will never contain all natural numbers.

@Carl: I fully agree. But a motive to write out a list or string of infinitely many digits without a finite formula is supplied by set theory: There it is claimed that uncountably many real numbers are definable by infinite digit sequences because all finite formulas can be shown to belong to a countable set.

Mathematics, restricted to this limitation would be very difficult, too digfficult. Therefore we have decided to apply every expression, with no limitation (in fact we can't, but the real needs are far below the limit). This is called potential infinity and is applied in real analysis very successfully. But then came Cantor with his drean of finished infinity, and that concept was doomed to failure. But it is such an attractive game for many, that they refuse to the the mistakes.

The explanation for the failure of set theory is simple to localize. Mathematics as far as it is discourse, dialogue, or monologue, is restricted to a finite number of bits - at most 10^100 if we could use the whole accessible universe. This restricts the Kolmogorov complexity of all expressions, for instance expressions defining numbers. It does not limit the values but their complexity. As an example take the 10-digit pocket calculator. It can display 10^80 but no number with 11 digits.

Every discussion suitable to show tits inconsistency is useful to keep newbies from being taken in by this absolutely useless theory. Here we have the chance to show that between all finite states with non-empty sets and the limit empty set there is a difference and hence soem transfer must have happened. Alas that can be excluded by the definition of the process. Contradiction.

@Carl: "if you deleted the whole off topic post, nothing would be lost." Unfortunalely this kind of supertasks is the basis of set theory. The original one, the story of Tristram Shandy, has been introduced by A. Fraenkel, a leading set theorist, in order to "explain" the enumeration of all fractions by natural numbers (see hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 81 and pp. 254ff).

RL is not realizable. Arbitrarily small balls are impossible because the diamter 10^-95 m is the lower limit (wavelength of a photon containing the whole energy of the universe). hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 353, Does the infinitely small exist in reality?

By the way, when in the urn problem always the last ball is removed, the "limit" is an infinite set of balls. In pyhsical reality however, the choice of labels would not change the result. Therefore the set-theoretical result is here, like everywhere else (see hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 124f, useless.

@amoeba: "do you also want to deny that the sequence of indicator functions $f_n$ converges to the constant zero?" Of course. It is the same procedure. At every finite step, it is increasing, but at infinity nothing can happen, according to the definition.

@aroth: You are completely right. By definition elements can only be transferred at finite stages. Never a decrease appears at finite stage. Therefore the snooty "cardinality is not continuous" is unjustified belief in miracles and violating the defined conditions. "Never" has no "in the end" or proper limit. Only the improper limit "infinity" is mathematically possible here.

You accept and apply the argument "all balls added before noon", but not all steps done before noon. Otherwise you would "understand" the phrase between all steps and the limit. It seems that you dislike to understand what could correct your mistaken views. Bye.

@Paul: "every numbered ball added to the urn eventually gets removed." Every numbered ball belongs to a tiny finite initial segment followed by infinitely many more. Every numbered ball is insufficient to conclude anything about a "final state". Why do you suppress these simple facts?

@Paul: Your section III discussion refers to "what happens at noon". Nothing happens at noon. It is impossible to describe this problem or Achilles and the tortoise by the sum of a geometric sequence. This approach fails. What happens is: the number of balls is potentially infinite. There is no last state or limit and no "all states before noon". Infinity cannot be finished.

@Paul: "The completed infinite series exists and its sum is 1" is a very naive an unmathematical opinion. 1 is not the term after all finite terms! There are no infinite sums defined in anylysis but only limits. The limit 1 has nothing to do with the existence of the complete sequence. But this mistaken view explains your confusion with sets.

@Paul: The limit of a sequence of real numbers is a real number. The limit of a sequence of sets is a set, here it is the empty set. This set is the contents of the urn, but not before any finite step. Therefore it is the contents after all finite steps. Hence we can conclude that between all finite steps and the limit something must have happened. The snooty "cardinality is not continuous" confirms just that. But according to the mathematical definition of the problem (with no reference to physics at all) nothing can have happened. Therefore we have a contradiction.

Completed infinities are not ubiquitous in mathematics (not in 0.999... because only the limit is 1) let alone in physics. Only set theory has them and requires in the current problem an exchange of balls between all finite steps and the allegedly existing limit, contrary to the definition of the problem. It is absolutely unphysical. Consider the sequence 1, 2, 3, ... of enumeretaed and returend balls, which has the limit empty set. But if always the same ball is used, we get the sequence 1, 1, 1, ... with limit 1, i.e., there is a ball in the urn. Result depends on labels? Not in physics!

cannot be soved by "the cardinality is not continuous". The cardinality is restricted to the definition of the problem: Only at finite steps transfer of balls is allowed.

@Paul: "If there is even a single ball in the limiting state" You describe the contradiction. The resolution is: There is no limiting state. But if it were, then the requirement to restrict transfer to finite steps would be not meaningless. Then there was the sequence n on the one side and the limit state on the other. Your mistake is to confuse seqeunces with Cauchy limits and sequences of sets. The sequence 0.9, 0.99, 0.999, ... has limit 1. But it does never reach 1. There is no term 1 in the sequence. The sequence of nsets however is said to evolve to the limit. That in contradicted and

@Paul: "So if you think there are somehow still infinitely many balls in the urn at noon, you must admit that not one of those balls can be a ball added before noon". You must admit that every ball n belongs to a tiny finite subset which is followed by an infinity of balls. The solution of the problem is that there is no completed infinity.

@Paul: It is no intuition but the definition of the problem that prohibits the empty limit: Transactions can only happen at finite steps, not between all finite steps and the limit. If, in the Ross-Littlewood case, the urn contains 9n balls at step n, then it cannot lose balls "between all finite steps and the limit". Therefore it cannot be empty in the limit.

@Aksakal: You have my full support! "Finsihed infinity" as only applied in Cantor's set theory, is a contradiction per se. It's never time to finally count the balls. The idea that "cardinality could be discontinuous" is not only weird but contradicted by the definition of the problem which requires that only at finite steps elements can be transferred.

"if no dollar remains forever, then the complete loss of all dollars leaves the empty set." No, you, uncounsciously perhaps, assume that there is an end. If the inflow never stops, then there is never an empty set in spite of every every dollar leaving.

The definition of the process restricts exchange of elements to finite steps. If all finite steps fail to get the empty set, then it is impossible to have the empty set as the limit. A theory yielding the empty set as limit is incompatible with the definition.

@Mario Carneiro: You claim that cardinality is not a continuous operation. But then you violate the mathematical premise that only at finite steps transfer may happen. If for all $n$, $f(n) = 9n$ and the final state is an empty urn, then between all $n$ and this finial state some transfer must have happened, contrary to the definition of the problem.

Perhaps you will be interested in a careful analysis of the problem given here: hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 252: The solution of McDuck's paradox. I would be interested in counter arguments.

@Mario Carneiro: The sequence $1/n$ has the limit 0. But this real number is not constructed by the terms of the sequence, It is defined by the formula. There is no necessity for an actual infinity of all terms of the sequence. Contrary to the set limit which is constructed by the terms of the sequence, namely by adding and removing balls. Unfortunately, as your answer shows, these very different premises are often confused. There is no physical intuition, but a math. definition: Every transfer of balls is bound to a finite step. Between all finite steps and the limit nothing can happen!

The sequence of pieces enumerated by {1}, {2}, {3}, ... has set limit { }. The sequence of pieces enumerated by {1}, {1}, {1}, ... has set limit {1 }. What happens if Donald gets coins??? Such a dependence on the labeling cannot satisfy scientific demands.

@Mario Carneiro: Set theory is not useful for a scientific problem (undistinguishable electrons fthat you mentioned). Here is the proof: Donald Duck daily earns one dollar and spends one dollar. As a cartoon character he can live forever and reach the "limit". What is this limit? If he gets banknotes enumerated by natural numbers, then in the limit he will be bankrupt. (See hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 55, example 1.) If he gets always the same banknote, say number 1, then in the limit (if existing) he will own this banknote.

@amoeba: "Nobody said that taking cardinality is a 'continuous' operation on sets" but by definition transfer of elements is only possible at finite steps. If for all finite steps the cardinality is not zero, then for all finite steps there are elements in the urn. An empty limit requires then that elements have been lost between all finite steps and omega. That is a contradiction. Or the limit has nothing to do with the problem.

Discontinuity is not the explanation. There is no set of all steps, there is no actual infinity. According to the discontinuity theory, at all finite steps there are balls in the urn and in the "limit" omega there are none. This is a contradiction since by definition transfers are only possible at finite steps. Between all finite steps and omega no transfer can happen.

@Luca Citi: You made no error. If, in the Ross-Littlewood case, the cardinality is f(n) = 9n, then the limit cannot be empty. By definition of the problem, transactions can only happen at finite steps. If the urn is not empty at all finite steps, it cannot be empty in the limit because between all finite steps and the limit no balls can be lost.

@Luca Citi: "We start with one ball and at each time the existing ball is removed and a new one is added, with probability one. The replacement takes no time. I think we all agree that (with probability one) there is exactly one ball at any time." Set theory yields the limit empty set. See hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 55, example 1.

@Mozibur Ullah: Countability is based on a bijection with |N. That requires that all elements of |N are "there". And between every two elements of the sequence we can interrupt and analyze what has been accomplished yet. Of course finished infinity is nonsense. If you want to learn the facts and how cranky set theorists act, you may do that best here: hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

@Mozibur Ullah: First; you are in error. Second: All techniques like induction do the same or can be forced to do the same since the sequence of natural numbers can be interrupted at every place or step. By the way, here you can see your error: hs-augsburg.de/~mueckenh/Transfinity/Hau.html

@mavavilj: Should you be one of the few lucky students who have a teacher not accepting Cantor's finished infinity? Of course countability is nonsense since the natural numbers are never complete because you can always find a greater number.

@Mozibur Ullah: To exhaust the natural numbers step by step is an insane approach. You will not be able to contradict this with insults.

@mavavilj: I think you are in error. The natural numbers are countable by definition. You may have been shown the uncountability of the real numbers. But there they are already starting to catch you and to pervert your brain. The notion of countability requires to complete or exhaust the infinite set of natural numbers. Only much later, after you will have been caught, they will offer you to believe that the natural can be exhausted step by step which is insane nonsense. But without warning you will fall prey to them. Observe how many will be "explaining this infinity" here.

@Mozibur Ullah : I know from thousands of intelligent laymen: formal mathematicians who teach counting to infinity step by step, are insane. Their teaching repels intelligent students and makes less intelligent students becoming insane cranks too.

@mavavilj: In some formal science (namely transfinite set theory) it is not possible to verify claims. My claim is that every natural number n has a successor n + 1. Therefore it is impossible to exhaust the set |N step by step and to arrive at infinity. Nevertheless the "formal mathematicians" cannot understand this, because they have acquired a brain damage which prevents them to recognize their mistake. - I see no other possibility. All healthy normal people can and do understand.

You may read about finished infinity. These ideas came from doctors of mathematics: "We repeat this process countably many times." [Paul J. Cohen: "The discovery of forcing", Rocky Mountain Journal of Mathematics 32,4 (2002) p. 1076f] "If this procedure is iterated aleph_2 times ..." [Karel Hrbacek, Thomas Jech: "Introduction to set theory", 2nd ed., Marcel Dekker, New York (1984) p. 232ff] It is hardly possible for PhDs in Pol Sciences to surpass something like that. A layman will recognize this at what it is.

@Bram28: Why are you perfectly comfortable? You simply accept it because they have taught you so. But if this were really justified, then the supertask would be as comfortable. The story of McDuck is a bijection with the natural numbers n and the fractions n/10. (From the first 10 fractions 1/10, 2/10, ..., 10/10 one is paired with the number 1.) The bankrupt is required because every fraction in McDuck's possession is not yet paired with a natural number.

@Bram28: You have learnt that bijections can be defined. But every infinite bijection can be understood and analyzed as a supertask. From that we find that the bijection requires strange things like an instance where McDuck is bankrupt. Most non-mathematicians do not believe in this bankrupt. But mathematicians, after having got explained this, have to believe in the bankrupt - unless they understand that infinite bijections contradict mathematics since the mathenmatical limit of the sequence (1/W(n)) = 0.