[FOM] Hazen on Liu Fengsui's infinite paradox

[Moses Klein]

Richard E. Grandy said:
> That resolves the paradox, but it still leaves something of a puzzle
> because we have a complete description of the state of the box up to
> the instant at the end of the minute, but that does not determine the
> state of the box at the important instant.  It looks as though this  may
> show the incompatability of (some forms of) determinism with
> infinite tasks of this kind.  (And I'm not going to try to say what
> "this kind" is this morning.)

I think the paradoxical nature comes from an implicit assumption that
functions will be continuous. You can read this one as a counterexample to
show that the cardinality function is not continuous. Let T_n be the set
of balls in the box at stage n; then (giving P(omega) the usual
Cantor-space topology), lim T_n is the empty set, but lim |T_n| is
infinite.

Like many paradoxes, I think this one can have a pedagogical use in
demonstrating the fallacy of assuming continuity. Beyond that, I don't see
much in it.

-----------
Moses Klein
Visiting Assistant Professor
Department of Mathematics
Ohio University
Athens, OH  45701,  USA
klein at math.ohiou.edu