Re: [Phys-L] Impossible?

[David Bowman <David_Bowman@georgetowncollege.edu>]

A key consideration in the Tarski-Banach Theorem here is that the sub-pieces -- although each continuously connected -- are *not* measurable sets (that cannot have any 'size' assigned to them) and, as such, they have an unimaginably tortuous 'shape' that I think can't even be assigned simple fractal/Hausedorff dimension.


-------- Original message --------
From: Chuck Britton via Phys-l <phys-l@mail.phys-l.org>
Date: 11/3/25 12:16 PM (GMT-05:00)
To: "Phys-L@phys-l.org" <Phys-L@Phys-L.org>
Cc: Chuck Britton <britton@ncssm.edu>, Donald Polvani <dgpolvani@gmail.com>
Subject: Re: [Phys-L] Impossible?

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The lesson is to not confuse mathematics with Reality.

This ‘Banach-Tarski’ paradox is a self consistent mathematical proof.

Unfortunately - it ignores the reality of atoms, molecules etc.

Mathematics is an extremely useful tool for Science and Engineering to use. But Tools must be understood to be useful.
Not all tools apply to Real problems.

> On Nov 3, 2025, at 11:20 AM, Donald Polvani via Phys-l <phys-l@mail.phys-l.org> wrote:
>
> From the NYT, 11/3/25
> "Two decades ago, the mathematician Moon Duchin spent her summers teaching geometry at Mathcamp, ... Another favorite was a famous “decomposability” paradox."
> “You can take a solid ball, cut it into five pieces that don’t overlap, move those pieces around and reconstruct two solid balls that are the same size as the one you started with,” Dr. Duchin, a professor of computer and data science at the University of Chicago, said in a recent interview." “That’s really weird, right?”
> Not only weird but impossible (it seems to me), since by a repeat process on each new solid ball an infinite amount of mass could be produced from a finite amount.
> Don Polvani
> Retired Adjunct Lecturer, Physics
> Anne Arundel Community  College
>
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