Modular Makeup
The Axiom of Choice can be a troublesome beast. It leads, for instance, to the amazing Banach-Tarski paradox, which states that one can cut up an orange and use the pieces to make two oranges. I found a wonderful book in the library on this topic, by Stan Wagon (here is the Google peek page).
A subset $U$ of a set $X$ is said to be paradoxical with respect to a group $G$ (used to rearrange pieces) if such a process can be done to $U$. The orange example comes from considering balls in $\mathbb{R}^{3}$ and translations and rotations, and it was shown by R. Robinson that only five pieces are needed to make a paradoxical orange!
Using the upper half plane and the modular group one can study similar paradoxes using Borel sets. Hausdorff showed, using the $S$ and $T$ presentation embedded in a rotation group, that the modular group is paradoxical. The relevant decomposition of hyperbolic space is three pieces $A$, $B$ and $C$ (see the pretty picture on the book cover) which are related via $TA = B$, $T^{2} A = C$ and $S A = B \cup C$.
What Tarski showed was that paradoxical decompositions are really about the non-existence of a finitely additive invariant measure.
A subset $U$ of a set $X$ is said to be paradoxical with respect to a group $G$ (used to rearrange pieces) if such a process can be done to $U$. The orange example comes from considering balls in $\mathbb{R}^{3}$ and translations and rotations, and it was shown by R. Robinson that only five pieces are needed to make a paradoxical orange!
Using the upper half plane and the modular group one can study similar paradoxes using Borel sets. Hausdorff showed, using the $S$ and $T$ presentation embedded in a rotation group, that the modular group is paradoxical. The relevant decomposition of hyperbolic space is three pieces $A$, $B$ and $C$ (see the pretty picture on the book cover) which are related via $TA = B$, $T^{2} A = C$ and $S A = B \cup C$.
What Tarski showed was that paradoxical decompositions are really about the non-existence of a finitely additive invariant measure.
posted by Kea | 3:44 PM
4 Comments:
Marni, I threw $15 at wordpress and redid the format for my .
Dang! That looked fine in "preview". You should dump blogger and move over to wordpress.
It is interesting that the congruence of open sets with non-empty interiors holds true only for D>2. In plane results are much weaker and area is same for congruent sets. Brings in mind 2-dimensionality of partonic 2-surfaces.
I would guess that the paradox disapperas if one can replace the axiom of choice with a weaker form restricting the choice to rationals or algebraics. This kind
of restriction of course makes sense only in a very special case when one has symmetries so that one can specify the preferred coordinates with respect to which the notion of rational point is defined.
Cool, Carl! I've made it my main link to you on my blogroll now. I've put some work into blogger and am a little reluctant to redo everything again at the moment. Maybe if I get a life one day.
I would guess that the paradox disappears if one can replace the axiom of choice ...
Hi Matti. At the end of the book, Wagon discusses the close relation between AC and the paradox: one can show that the paradox is unprovable in ZF alone, so it really hinges on AC. In the topos Set, AC is a simple condition but not one of the elementary axioms, so it is easy enough in higher topos theory to work without AC. I like this idea for many reasons.
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