FOM: Paradoxical decompositions of space

[Stephen G Simpson]

Solomon Feferman writes:
 > Let n be at least 3 and A and B be nonempty bounded subsets of R^n
 > with nonempty interior.  Then there is a pairwise disjoint
 > collection {A1,...Ak} of open subsets of A whose union is dense in
 > A and a pairwise disjoint collection {B1,...Bk} of open subsets of
 > B whose union is dense in B such that each Ai is isometric with Bi.

This is a very interesting gloss on the comments of David Ross, John
Baldwin, and John Steel.

Obviously A - (A1 union ... union Ak) is of positive measure in the
interesting cases.  I guess the source of the "pathology" here is
traceable to the following familiar example: For any epsilon > 0 we
can easily define an open set of reals of measure epsilon, call it S,
which covers the rationals and is therefore dense in the reals.  This
construction also doesn't use the axiom of choice.

 > Is this a sign of deep pathology infecting set theory, or is it
 > just a sign that one should not rely on one's intuitions when it
 > comes to the concepts involved, or what...?

The second, I would say.  The example S above is frequently used to
illustrate how paradoxes can help us to sharpen our understanding of
various concepts.

The paradox mentioned by Feferman seems different in kind from the
Banach-Tarski paradox, because the latter "pathology" can't happen for
a very wide class of sets, the Lebesgue measurable sets.  I guess
Harvey's neo-relativism is relevant here.  It's easy for me to imagine
mathematics in a universe excluding non-measurable sets, but it's hard
for me to imagine mathematics in a universe excluding S.

-- Steve