[FOM] Hazen on Liu Fengsui's infinite paradox
Moses Klein
klein at math.ohiou.edu
Fri Nov 22 13:26:28 EST 2002
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
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