[FOM] Hazen on Liu Fengsui's infinite paradox

Richard E. Grandy rgrandy at rice.edu
Fri Nov 22 09:05:47 EST 2002

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.)

Richard Grandy
Rice University
Houston TX USA

>   Liu Fengsui presented a paradoxical scenario, involving the movement of
>infinitely many balls in and out of urns, with plausible arguments to show
>that one urn both would and would not be empty at the end of the process,
>and asks what is known about it.
>   I believe the scenario he presents is known in the literature as "Ross's
>Paradox."  It is one of the family of puzzles known generically as
>"supertasks": processes (like that of Zeno's runner running through the
>infinitely many subintervals of an interval) consisting of infinitely many
>successive parts.
>   A good reference on this family of paradoxes is
>	John Earman and John D. Norton, "Infinite pains: the trouble with
>supertasks," in Adam Morton & Stephen P. Stich, eds., "Benacerraf and his
>Critics" (Oxford: Blackwell, 1996; ISBN0-631-19268-9),
>which discusses Ross's Paradox among others.
>   The usual resolution is to is to say that the description of the scenario
>only implies that the the urn contains a positive number of balls after
>each finite number of transfers, and so allows it to be empty at the end of
>infinitely many.  It is thus analogous to "The Thompson Lamp," another
>standard supertask.  The lamp is turned on after 30 seconds, off afte 15
>seconds more, back on after another 7.5, and sdo on, infinitely many times.
>Paradoxical conclusion: at the end of the minute is both on (because it was
>turned back on after every time at which it was turned off) and off
>(because it was turned back off after every time at which it was turned
>on).  Resolution: the initial description determines the state of the lamp
>at each instant during the minute, but doesn't imply anything about whether
>it is on or off when the minute is over.
>Allen Hazen
>Philosophy Department
>University of Melbourne
>FOM mailing list
>FOM at cs.nyu.edu

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