In the exchange paradox, two players receive envelopes containing different amounts of money. The assignment of the amounts ensures each player has the same probability of receiving each possible amount. Nonetheless, for each specific amount a player may find in his envelope, there is a positive expectation of gain if the player swaps envelopes with the other player, in apparent contradiction with the symmetry of the game. I consider a variant form of the paradox that avoids problems with improper probabilities and I argue that in it these expectations give no grounds for a decision to swap since that decision must be based on a summation of all the expectations. But this sum yields a non-convergent series that has no meaningful value. The conflicting recommendations - that it is to one or the other player's advantage to swap - arise from different ways of grouping terms in the sum that yield an illusion of convergence. I describe a generalized exchange paradox, explore some of its properties and display another example. © 1998 University of Southern California and Blackwell Publishers Ltd.

We discuss two supertasks invented recently by Laraudogoitia [1996, 1997], Both involve an infinite number of particle collisions within a finite amount of time and both compromise determinism. We point out that the sources of the indeterminism are rather different in the two cases - one involves unbounded particle velocities, the other involves particles with no lower bound to their sizes - and consequently that the implications for determinism are rather different - one form of indeterminism affects Newtonian but not relativistic physics, while the other form is insensitive to the classical vs relativistic distinction. We also note some interesting linkages among supertasks, indeterminism and foundations problems in the general theory of relativity.