Suppose that a point is uniformly chosen on the circumference of the circle *T*. Write *A*≤*B* for "the point is at least as likely to be in *B* as in *A*" and say *A*<*B* when *A*≤*B* but not *B*≤*A*. Here are some very plausible axioms:

- If
*A*≤*B*and*B*≤*C*, then*A*≤*C*. (Transitivity) -
*A*≤*A*. (Reflexivity) - Either
*A*≤*B*or*B*≤*A*(or both). (Totality) - If
*A*is a proper subset of*B*, then*A*<*B*. (Regularity)

- If
*r*is any reflection in a line going through the center of the circle*T*and*A*≤*B*, then*r**A*≤*r**B*,

*A*and

*B*if and only if it holds between their reflections.

**Proposition.** There is no relation ≤ satisfying (1)-(5) for all countable subsets *A*, *B* and *C* of *T*.

I do not as yet know if the Proposition is true if we replace reflections by rotations in (5).

Totality and/or Regularity should go. Other cases suggest to me that both should go.

**Proof of Proposition:** Suppose ≤ satisfies (1)-(5). Say that *A*~*B* if and only if *A*≤*B* and *B*≤*A*. It is easy to see that ~ is transitive since ≤ is transitive and if *A*~*B* then *r**A*~*r**B*. Now observe that *r**A*~*A*. For either *r**A*≤*A* or *A*≤*r**A* by totality. If *r**A*≤*A* then *A*=*r*^{2}*A*≤*r**A* (the square of a reflection is the identity). If *A*≤*r**A* then *r**A*≤*r*^{2}*A*=*A*. In both cases, thus, *A*~*r**A*.

Therefore, if *A*≤*B*, then *r**A*~*A*≤*B* and so *r**A*≤*B*. Now, any rotation can be written as the composition of a pair of reflections (a rotation by angle θ equals the composition of reflections in lines subtending angle θ/2). Thus, for every every *rotation* *r*, if we have *A*≤*B*, then we have *r**A*≤*B* and *r**A*≤*r**B*. It follows easily that *A*<*B* if and only if *r**A*<*B*.

Now, let *r* be a rotation by an angle which is an irrational number of degrees and let *x*_{0} be any point on the circle. Let *A* be the set {*x*_{0},*r**x*_{0},*r*^{2}*x*_{0},*r*^{3}*x*_{0},...}. Observe that *r**A*={*r**x*_{0},*r*^{2}*x*_{0},*r*^{3}*x*_{0},*r*^{4}*x*_{0},...} is a proper subset of *A* (*x*_{0} is not equal to *r*^{n}*x*_{0} for any positive integer *n* as *r* was a rotation by an irrational number of degrees). Thus, *r**A*<*A* by Regularity. Thus, *A*<*A*, which is a contradiction.

## 1 comment:

The answer to the question after the Proposition is negative: http://arxiv.org/abs/1309.7295

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