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Mathematical notes on Ross's paradox.

Allis and Koetsier [1991] have recently discussed what they call 'Ross's Paradox'. With trivial terminalogical modifications, the procedure that gives rise to it is as follows. We have an infinitely large urn, and an infinite set of balls corresponding to and labelled by the natural numbers. At time t = 0 we put balls 1 and 2 into the urn and remove ball 1. At time t = 1/2 we put in balls 3 and 4, and remove ball 2. Continuing in this way, at time t = 1 - [(1/2).sup.n] we put in balls 2n + 1 and 2n + 2, and remove ball n + 1. What is the content of the urn at time t = 1? The 'paradox' is the contradiction between the fact that the content of the urn is always increasing; and an argument that the urn is empty at time t = 1. Consider ball j. It was put into the urn at time t = 1 - [(1/2).sup.[1/2(j-1)]], where [x] denotes the integral part of x, and removed at time t = 1 - [(1/2).sup.j-1]. At time t = 1 therefore, for each j, ball j is not in the urn, which must therefore be empty. The 'common sense' answer, an infinite content, is rebutted by asking for the label of even one of the balls in the box at t = 1. To avoid performing two operations simultaneously that might interfere with each other, the removal of ball n + 1 can be deferred to time t= 1 - [(1/2).sup.n] + [(1/2).sup.n+2].

The paradox involves several points of first year University level mathematical analysis, and I hope they were raised by the mathematics students at the Free University of Amsterdam, mentioned in the article. They include the definition of functions, the mathematical representation of a physical state, the concept of convergence and its preservation under mapping, and the rearrangeability of infinite series.

One aspect of the problem is that the content of the urn at time t, which I will denote by [X.sub.t], is defined explicitly only at the set of values t = 1 - [(1/2).sup.n] for integral n [is greater than or equal to] 1, (which will be denoted by [t.sub.n]), at which instants [[X.sub.[t.sub.n]] = n + 1. We simply do not know how many balls are in the urn at time 1, nor indeed at any instant other than those specified. It is possible to define [X.sub.t] quite arbitrarily for other values of t, and there are many acceptable ways of doing so. We can therefore try to answer the question 'How would a reasonable person complete the definition of [X.sub.t]?' This is the issue to which Allis and Koetsier devote most of their discussion. The most natural definition is, as they point out, (i) [X.sub.t] = [[X.sub.[t.sub.n]], when 1 - [(1/2).sup.n] [is less than or equal to] t [is less than] 1 - [(1/2).sup.n+1]. This represents persistence of the content of the urn until a change is positively specified. If [X.sub.t] did not have to be integral for values of t other than 1 - [(1/2).sup.n] we could try the linear interpolation (ii) [X.sub.t] = n + 1 + [2.sup.n+1](t - [t.sub.n]), or the smoother nonlinear interpolation (iii) [X.sub.t] = -[log.sub.2] (1 - t). An alternative, (iv) [X.sub.t] = 0, when t [is not equal to] [t.sub.n] for some integer n [is greater than] 0, may seem a little harder to justify, but represents the idea that the urn is used as a mixing bowl, and that in between activities involving it, the balls are stored in some other container.

A point that comes out very strongly in the original discussion is that since the function is defined for the sequence 1 - [(1/2).sup.n], the domain to be covered by any extension should be the open set {x: 0 [is less than or equal to] x [is less than] 1}, not the closed set {x: 0 [is less than or equal to] x [is less than or equal to] 1} and that the definition of the function x = 1 is a problem of a different nature from that of its definition in the intervals between values of 1 - [(1/2).sup.n]. It is however natural to want to complete the definition by setting [X.sub.1] = [limit of] [X.sub.t] as t approaches 1. According to (i), (ii) and (iii) we have [X.sub.t] approaches [infinity] as t approaches 1, and according to (iv) there is no limit, but 0 is a cluster point.

Another aspect of the paradox lies in the distinguishability or otherwise of the balls. It is not important that they should be labelled before the start of the process. Those that are put into the urn at time 1 - [(1/2).sup.n] can be made to acquire thereby the labels 2n, 2n + 1. If the balls are unlabelled, the natural way of recording the number in the urn at any time is a single (i.e., 1-dimensional) number. If they are labelled, and hence distinguishable, the full specification of the content of the urn requires an infinite dimensional vector, whose j-th component takes the values 1 or 0, according to whether ball j is or is not in the urn. When t lies in the interval 1 - [(1/2).sup.n] [is less than or equal to] t [is less than] 1 - [(1/2).sub.n+1], the vector has 0 in its first n positions, 1 in positions n + 1 to 2n + 2, and 0 the remaining positions. Such infinite dimensional vectors form a space in which the convergence of the sequence in which we are interested will depend on the topology. Now a function from one topological space to another, which may be elementwise identical to the first but equipped with a different topology, may or may not preserve convergence properties. It is said to be continuous if it does. If we define a sequence of infinite vectors to be convergent if the sequence of j-th components converges for every j, then the sequence that represents the evolution of the contents of the urn converges to the infinite dimensional zero vector. This is in agreement with the argument that the urn is empty at time t = 1. If on the other hand we were to use the topology based on the metric [[absolute value of] d.sup.p] = [[Sigma].sub.j] [[Sigma].sub.j][[absolute value of] [d.sub.j].sup.p], where [d.sub.j] is the difference between the j-th components of the vectors and d the consequent distance between the vectors, then the sequence diverges. In particular the choice p = 1 corresponds to judging the convergence or otherwise of the set of vectors by the corresponding property of the sequence of cardinals of the content of the urn. The function from the infinite dimensional vectors with the topology of pointwise convergence to the sum of moduli of their components is discontinuous, its type of discontinuity being nearer to the intuitive idea of unboundedness than to that of 'jumps' in the graph. A major source of the paradox lies in judging the convergence of the sequence of contents of the urn simultaneously in terms of two incompatible concepts of convergence.

The mutual relations of positive and negative terms involved in the addition to and subtraction from the urn seems possibly relevant to the paradox. Consider the series expansion of the complex function 1/(1 - z). On the open disk {z: [absolute value of] x [is less than] 1} this is given by the series

[summation of] [z.sup.n] where n = 0 to [infinity]

which converges absolutely and uniformly. Now take [z.sub.0] = - 1 and look at the sequence

[summation of] [(- 1).sup.n] where n = 0 to [infinity]

in the spirit of Ross's paradox. It is not convergent, because the partial sums alternate between the values 1 and 0. However 'common sense' would suggest that if we were to 'find' a sum for the series, it should be 1/(1 - (- 1)) = 1/2. This can be achieved by a method of general applicability and interest, if we replace the values of the alternating partial sums by the sequence formed by averaging the first k terms, for k = 1, 2,.... The resulting series converges to 1/2, and this is called the 'Cesaro sum' of the original series. The sequence of contents of the urn can be described, particularly if we take the modification in the last sentence of para. 1, by the partial sums of the sequence 2 - 1 + 2 - 1 + 2 - 1 + ... Because the positive and negative parts of this sequence both diverge, it can be rearranged so that the partial sums diverge to +[infinity] or -[infinity], or oscillate finitely or infinitely. However in this case any reasonable summation method would produce a transformed sequence that would diverge to +[infinity], and leave the identification of the value at time t = 1 with the limit as the only outstanding problem.

The problem can be generalised by asking what happens if we continue the process beyond time t = 1. Since we have used up the balls labelled with finite integers, the balls that we put in at time t = 1 will have to be labelled [Omega] + 1 and [Omega] + 2, where [Omega] is the first infinite ordinal, and we remove number [Omega] + 1. At time 1 1/2 we put in balls [Omega] + 3, [Omega] + 4 and remove [Omega] + 2, etc. Now glance back to time t = 1. Is the content of the urn [Mathematical Expression Omitted], the cardinal corresponding to [Omega]?

PHILIP HOLGATE Department of Mathematics and Statistics Birkbeck College, Malet Street London WC1E 7HX


ALLIS, V. and KOETSIER, T. [1991]: On some paradoxes of the infinite. British Journal for the Philosophy of Science, 42, pp. 187-94.
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Title Annotation:response to V. Allis and T. Koetsier, British Journal for the Philosophy of Science, vol. 42, p. 187, 1991; paradoxes of the infinite
Author:Holgate, Philip
Publication:The British Journal for the Philosophy of Science
Date:Mar 1, 1994
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