FOM: Paradoxes, including Banach-Tarski ...
John Case
case at eecis.udel.edu
Sat Jan 3 12:36:06 EST 1998
RE: Paradoxes in general and some initial ideas re Banach-Tarski
I'll exclude from the remaining discussion paradoxes which depend on making a
mistake in the inference mechanism for obtaining the paradoxical conclusion,
e.g., I'll not discuss pedagogically useful incorrect proofs of oddball
conclusions obtained by improperly applying math induction --- leaving out the
base cases or something slightly more interesting.
Roughly a paradox is ~~ 1/2 of a proof by contradiction of the opposite of one
of the premises from the other premises, where some of the premises are
typically hidden and the contradiction may only thereby be felt, not explicitly
derived. It's the part of such a proof where one (almost or actually) derives
the contradiction from the larger set of premises. It's not the part where one
derives the opposite of one of the premises from the other premises. That part
is _a_ resolution of the paradox. For example, one resolution of the liar
paradox is Tarski's Theorem (first known to Goedel). N.B. Sometimes the
original paradox is informal and its resolutions require filling in more than
the hidden premises. For example, I vaguely recall that Myhill considered an
interesting negationless, interpreted language in which one _could_ express a
truth definition. However, filling in the language part with first-order
arithmetic, ... instead, yields Tarski's Theorem: ~~ the language can_not_
express its standard truth definition.
Some popular literature is content to mystery monger about paradoxes endlessly,
e.g., about various annoying and, especially, annoyingly insuccinct
versions of the liar paradox --- without drawing any substantive
conclusions or resolutions. As essentially noted by Harvey and emphasized by
Martin Davis, we learn something (sometimes something profound) when we
_resolve_ a paradox: fill in the missing parts (which language, ...), tease
out the hidden premises, decide which premise to take the opposite, and derive
that opposite from the remaining premises. Then we get a theorem, some
possibly new, important, and profound information.
Lets see how this might apply to Banach-Tarski. Steve Simpson nicely suggested
a resolution. Negate the premise that our worlds of sets allow non-Borel
stuff. I was not personally in favor of negating that premise, but already
said that I thought studying the resultant world of sets is interesting. I
personally favor allowing choice to stay in the set of premises. For me the
paradoxical element is not non-measurable sets, for me it's the _possibility_
that _real physical_ space might admit of such a decomposition-recomposition.
My gut reaction is that real physical space does not admit of such a
restructuring. Hence, the first, unrefined resolution of Banach-Tarski that
occurs to me is that real physical space is not one of the spaces for which we
get Banach-Tarski. That would leave questions like Which mathematical
space more accurately models real physical space? Hmmm, for the experts: Do
relativistic spaces have a Banach-Tarski paradox? Is there some notion of
rigid there permitting Banach-Tarski (in some form) _and_ in a form that
makes us feel (if not explicity derive) that something is wrong?
Another unrefined resolution occurs to me. This time we negate my premise that
real physical space does not admit Banach-Tarski restructuring. We get the
possible conclusion that _real physical_ space does. From definibility results
Harvey alluded to, one suspects we might have trouble doing such a
restructuring on purpose in the lab. But it might, with low probability,
happen spontaneously? I'm thinking very vaguely here about stuff like quantum
fluctuations that some cosmologists believe might have brought our free lunch,
the universe, into existence in the first place.
The above considerations re Banach-Tarski come under the space part of
Harvey's
1. Mathematics starts with counting and measuring. These matters are
closely connected with modeling various physical phenomenon involving space
and time.
I'd like to see some more resolutions to believe in from the experts.
(-8 John Case
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