From: Eliezer S. Yudkowsky (email@example.com)
Date: Mon Feb 04 2002 - 13:39:06 MST
> The Banach-Tarski 'decomposition' states that it is possible to dissect a ball
> into (six) pieces which can be reassembled by rigid motions to form two
> balls of the same size as the original. At first glance this seems to contradict
> some of our intuition about physics. Consequently this decomposition is often
> called the Banach-Tarski 'paradox'. But, actually, it only yields some complication
> to our knowledge of this word: ' volume ' .
> [But - who knows ? - could we, one day, n-plicate this universe?]
Saying that you can "dissect" a ball into six "pieces" is a bit
misleading. What they would call a "piece", we would call "an arbitrary
set of real points that moves by translation and rotation". In this case,
the term "arbitrary set" is to be taken literally; the sets can only be
constructed using the Axiom of Choice.
The term "dissect" is also misleading. The pieces here cannot be created
by any number of cuts, no matter how large. The sets of points do not
have analytic boundaries. You can't create them by "cutting", even with a
Furthermore, you cannot cut up any object composed of a finite number of
atoms. This isn't, as sometimes stated, because atoms are "too large".
Even if the object were composed of an infinite but countable number of
atoms, it wouldn't be enough.
-- -- -- -- --
Eliezer S. Yudkowsky http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence
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