The Ross-Littlewood Paradox

Ross-Littlewood Paradox [Image from Steemit website: here. ]

A most perplexing paradox appeared in Littlewood’s book A Mathematician’s Miscellany. It was later analysed in detail by Sheldon Ross in his 1988 book A First Course in Probability.

Littlewood wrote:

Balls numbered 1, 2, … (or for a mathematician the numbers themselves) are put into a box as follows. At 1 minute to noon the numbers 1 to 10 are put in, and the number 1 is taken out. At half a minute to noon numbers 11 to 20 are put in and the number 2 is taken out. At one quarter minute before noon 21 to 30 in and 3 out; and so on. How many are in the box at noon? The answer is none: any selected number, e.g. 100, is absent, having been taken out at the 100th operation.

(Littlewood, 1953, Page 5).

Simplified Version of the Paradox

We start with an infinite number of balls, numbered sequentially. At each stage, two balls are placed in an urn and then one is removed and discarded. The stages are telescoped so that they are completed in a finite time ${T}$ . How many balls are in the urn after time ${T}$ ? We examine two strategies.

Strategy I: Remove the first member of each pair

We tabulate the balls in the urn, and their number, at each stage:

Ross-Littlewood-Table1 After stage n, there are n balls in the urn. Ultimately, all the odd-numbered balls have been discarded. Therefore, all the even-numbered balls remain in the box.

Strategy II: Remove the ${n}$ -th ball at stage ${n}$

Now we consider Littlewood’s strategy: we remove the ${n}$ -th ball at stage ${n}$ .

Ross-Littlewood-Table2

Apparently, there are still ${n}$ balls in the urn after the ${n}$ -th stage. However, the numbers ${\{1, 2, \dots , n\}}$ are missing. Thus, at the completion of the process, balls with every natural number have been removed! We must conclude, as Littlewood did, that ultimately, there are no balls left in the box!

Supertasks

Supertasks were first suggested by Hermann Weyl although he did not use that term. A supertask is a countably infinite sequence of operations that occur sequentially in a finite interval of time. The term was coined by the philosopher James F. Thomson, who devised the paradox of Thomson’s lamp.

One obvious resolution of the Ross-Littlewood paradox is to say that supertasks are impossible. Thomson emphatically denied that supertasks are possible. If they were possible, propositions in number theory, such as Goldbach’s conjecture, could be determined in a finite time by a search of the set of natural numbers ${\mathbb{N}}$ .

Sources

${\bullet}$ Littlewood, John E., 1953: A Mathematician’s Miscellany. Methuen & Co.~Ltd., London. Reissued as Littlewood’s Miscellany, Be’la Bolloba’s (ed.), Cambridge University Press, (1986). ISBN: 0-521-33058-0.

${\bullet}$ Ross, Sheldon, 1988: A First Course in Probability. Ninth Edn 2012. Pearson Publ. ISBN: 9780-3217-9477-2.

${\bullet}$ Thomson, James F., 1954: Tasks and Super-Tasks. Analysis, 15(1), 1–13. doi: 10.2307/3326643.

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