In the paper below the authors describe three super-tasks. They show that although the abstract notion of a super-task may be, as Benacerraf suggested, a conceptual mismatch, the completion of the three super-tasks involved can be defined rather naturally, without leading to inconsistency, by means of a particular kinematical interpretation combined with a principle of continuity.

In an earlier paper the authors discussed some super-tasks by means of a kinematical interpretation. In the present paper we show a semi-formal way that a more abstract treatment is possible. The core idea of our approach is simple: if a super-task can be considered as a union of (finite) tasks, it is natural to define the effect of the super-task as the union of the effects of the finite tasks it consists of. We show that this approach enables us (...) to handle two of the three super-tasks that we discussed earlier. We also argue that recent objections against our original kinematical interpretation do not hold water. One of our arguments is based on the construction of an elegant correspondence between the first of those three super-tasks and Zeno's Achilles and the Tortois. (shrink)

We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.