FOM: Paradoxical decompositions of space
Solomon Feferman
sf at Csli.Stanford.EDU
Sun Dec 14 11:54:57 EST 1997
In a paper entitled "Banach-Tarski paradox using pieces with the property
of Baire", Proc. Natl. Acad. Sci. USA 89 (1992) 10726-10728, the authors
prove not only the refinement of BT implicit in the title but also the
following theorem that I will denote OBT ('O' for opens).
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.
OBT is proved *without* the Axiom of Choice, but does use the
group-existence argument which is the first step in the Hausdorff and
Banach-Tarski proofs. This seems to underscore what John Ross has been
stressing in his postings on BT. At any rate, I'd like to hear what
people think of OBT and how it might affect what we think of restricting
to the mathematics of Borel sets. 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...?
(Guess where I might come out on this.)
A technical question is what part of ZF is needed for OBT. Does Simpson's
TBU_0 suffice?
Sol Feferman
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