On some paradoxes of the infinite II.

1 Background

Paradoxes of the infinite are deceptive research topics: often we are led astray by intuitions and reasoning patterns from which the flawed ingredients can only be extracted through thorough investigation.

A particularly interesting paradox, regarding infinitely many balls and an urn, has been described in Allis and Koetsier [1991]. There we have described three super-tasks, each a variant on the same theme. Through a kinematical interpretation, and application of a continuity principle, we were able to solve the paradoxes involved.

Recently, two interesting reactions to our paper appeared. Holgate [1994] added some mathematical remarks to our paper. However, van Bendegem [1994] has argued that at least two of the three super-tasks described in our 1991 article are impossible. Those two papers prompted us to discuss the concept of super-tasks, and our kinematical interpretation in a more abstract fashion. Through these discussions we have formulated a more formal interpretation and an abstract continuity principle for super-tasks, which is the topic of this article.

We believe that the abstract continuity principle, which is an extension of, and consistent with our original kinematical continuity principle, leads to acceptable results for many super-tasks.

2 The Littlewood-Ross paradox

In Allis and Koetsier [1991] we have described three super-tasks, of which we attributed the first to Ross ([1988], p. 46), while the other two are variants proposed by the authors. Recently, we have found that what we called Ross' Paradox is described already in Littlewood [1953], although without critical discussion. From now on, we will refer to it as the Littlewood-Ross paradox.

The following formulation of the super-task leading to Littlewood-Ross paradox is taken from Allis and Koetsier [1991].

Suppose that we possess an infinitely large urn and an infinite collection of balls, labeled 1, 2, 3 and so on. There are exactly as many balls as there are natural (standard) numbers. Consider the following thought experiment. At 1 minute to 12 p.m. balls numbered 1 through 10 are placed in the urn, and ball number 1 is withdrawn; at 1/2 minute to 12 p.m., balls numbered 11 through 20 are placed in the urn, and ball number 2 is withdrawn; at 1/4 minute to 12 p.m., balls numbered 21 to 30 are placed in the urn and ball number 3 is withdrawn; at 1/8 minute to 12 p.m., balls numbered 31 to 40 are placed in the urn, and ball number 4 is withdrawn, and so on. How many balls are in the urn at 12p.m., and what are their labels?

The paradox is based on the existence of two arguments leading to different conclusions. The first argument observes that at each step the number of balls in the urn increases with 9. Thus, at 12 p.m., after infinitely many steps, the urn must contain infinitely many balls. The second argument focuses on individual balls: ball n enters the urn at some time before 12 p.m. and leaves the urn at step n (i.e. at [1.sup.n-1/2] minute to 12 p.m.). Thus, as each ball leaves the urn before 12 p.m., the urn must be empty.

The Littlewood-Ross paradox gives rise to at least three different types of reactions. First, constructivists in the philosophy of mathematics and others who reject the actual infinite dismiss super-tasks as objects of investigation. Second, some, such as van Bendegem [1994], only accept super-tasks it they can be performed under constraints imposed by the real world, such as upper bounds on speed and acceleration. Third, some, such as the authors, investigate super-tasks from a non-constructivist logico-mathematical point of view.

In Section 3 we formally define tasks and super-tasks and the relation between the two. We will show that the distinction between (finite) tasks and super-tasks is not as clear as might be expected, rendering their investigation possibly even interesting to the first type of people mentioned above. The abstract continuity principle is described and applied in Section 4. As will become clear, our formal interpretation leads to an empty urn at 12 p.m. in case of the Littlewood-Ross thought experiment, but to an urn with infinitely many balls in a closely related variant of the experiment. In Section 5 we show that there exists a correspondence between the Littlewood-Ross super-task and Zeno's Achilles and the Tortoise. In Section 6 we discuss van Bendegem's (1994) paper. We will show that the Littlewood-Ross super-task can be performed under some restrictions imposed by the real world, contradicting van Bendegem's conclusions.

3 Tasks and super-tasks

Informally, the difference between a task and a super-task is that a task …