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.

4 Responses so far »

  1. 1

    CarlBrannen said,

    Marni, I threw $15 at wordpress and redid the format for my .

  2. 2

    CarlBrannen said,

    Dang! That looked fine in “preview”. You should dump blogger and move over to wordpress.

  3. 3

    Matti Pitkanen said,

    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.

  4. 4

    Kea said,

    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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