Re: Questions about Axiom of Choice

In article <1157473528.401566.308170@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<agapito6314@xxxxxxx> wrote:

[.re The Banach-Tarski paradox.]


Thank you and everyone who replied. This is a really dumb question but
is it not possible to prove or disprove this physically?

No. The "cuts" in question cannot be performed in reality, or at least
they cannot be performed the way we understand reality. Remember
that geometrically we consider the space to be infinitely divisible,
but our understanding of reality does not allow infinite divisibility;
we cannot go anywhere smaller than the Planck length. As such, it
would be even theoretically impossible to perform the actual "cuts" of
the unit sphere, even setting aside the problem of errors of
measurement which would certainly creep in ('theoretically impossible'
at least within the framework of our current understanding of matter).

In addition, the "cuts" are fairly complicated affairs; if you've ever
seen a construction of the Vitali set (the typical example of a
non-measurable subset of the real line) you will probably realize that
it would be impossible to "physicially" represent this set. The same
occurs with the slices needed in the Banach-Tarski paradox; we can
prove they exist, but we cannot really ->describe<- them in any way
that would allow you to even theoretically attempt to perform them
(even setting aside the physical constraints above).

That is we
can construct a solid unit radius ball, cut it up into a finite number
of pieces (according to some procedure) and produce the new (solid?)
bigger ball? Thanks again.

Surely someone would have done that with a nice little ball of gold by
now... (-:

Alas, space is not infinitely divisible.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

Relevant Pages

  • Models and Theories ( was Evolution increases the computational ability of organisms.)
    ... Models are NOT derived from "physical reality". ... Zeno's Paradox which is essentially the same problem as Hamilton's Rule: ... an empirical paradox (a paradox of science). ... Retrograde motion of the planets was explained by ...
    (sci.bio.evolution)
  • Re: Pious untruth
    ... harps upon in reaction against some ... Rearranging values and meanings is not what I ... This paradox is a lot like Zeno's paradoxes of motion, ... Zeno, according to most reports, was concerned with highlighting the logical paradoxes inherent in various assertions regarding reality - specifically that reality was multiple and changing, versus One and unchanging. ...
    (talk.religion.buddhism)
  • Re: Pious untruth
    ... harps upon in reaction against some ... The reality that we receive in sensation (what you refer to as <<the ... Rearranging values and meanings is not what I ... This paradox is a lot like Zeno's paradoxes of motion, ...
    (talk.religion.buddhism)
  • Re: Pious untruth
    ... harps upon in reaction against some ... The reality that we receive in sensation (what you refer to as <<the ... Rearranging values and meanings is not what I ... This paradox is a lot like Zeno's paradoxes of motion, ...
    (talk.religion.buddhism)
  • Re: Pious untruth
    ... harps upon in reaction against some ... The reality that we receive in sensation (what you refer to as <<the ... Rearranging values and meanings is not what I ... This paradox is a lot like Zeno's paradoxes of motion, ...
    (talk.religion.buddhism)
Loading