Let X be the set of integers, or a circle, or the real line, or Euclidean n dimensional space. Imagine a point is "uniformly" randomly chosen in X. For any two subsets A and B of X, we would like to be able to say if one of the subsets is more probable as the location of the point. Here are some conditions we want to impose on the ≤ comparison:
- ≤ is a total preorder: for any A, B and C, we have A≤B or B≤A; A≤B and B≤C implies A≤C; and A≤A.
- For any translation t of X (where we deem rotations on the circle to count as "circular translations"), we have tA≤tB if and only if A≤B. (≤ is translation-invariant)
- If A is a proper subset of B, then A<B (i.e., A≤B but not B≤A).
- If m(A)<m(B) for d-dimensional Hausdorff measure (including of course Lebesgue measure), for any d between 0 and the dimension of the space (inclusive), then A<B.
Proposition 1. Given the Axiom of Choice, there is an ordering ≤ satisfying 1-4.
Proof: Start with an ordering such that A is less than B if and only if A=B, or m(A)<m(B) for any d-dimensional Hausdorff measure, or A is a proper subset of B. This ordering is translation-invariant, and it extends to a preorder satisfying 1-4 by the main theorem of Section 2 here.
That sounds great! We can finally compare probabilities of landing in arbitrary sets, it seems. Well, almost. Given a uniform distribution, we would at least want the invariance also to hold for coordinate reflections (where we reflect the kth coordinate, for any k).
Proposition 2. There is no ordering ≤ satisfying 1-3 and the coordinate reflection condition.
That's a consequence of the final proposition in the paper I linked to above.
What a surprising difference these reflections make! With just translations, we have a lovely invariant order (though presumably not unique) respecting strict inclusions of sets. When we add coordinate reflections, we don't. Technically, the difference is that once we have reflections and translations, our symmetry group is no longer commutative. And of course, in the Euclidean space case if we add rotations, all is lost, too (that, too, is easy to show).
Philosophical corollary. There can be incommensurably probable events, and hence incommensurably valuable events (since two chances at the same good will be incommensurably good if the chances are incommensurably probable).
7 comments:
This may be a bit off-topic, but why is it that cardinality is defined in terms of ability to be put in a one-to-one correspondence? Why didn't mathematicians define cardinality as:
"Two sets have the same cardinality iff they can be put in a one-to-one correspondence (bijective) and cannot be paired in an injective non-surjective or non-injective surjective way."
This gets rid of the messy problems with transfinite cardinaliy.
"This gets rid of the messy problems with transfinite cardinaliy."
Maybe, alas the cost is that given the Axiom of Choice, then there are no transfinite cardinalities any more: An infinite set will no longer even have the same cardinality as itself!
(Without the Axiom of Choice, this isn't exactly true. What is true is that then a set that contains a countably infinite set will no longer have the same cardinality as itself. But without something like at least Countable Choice, one can't prove that every infinite set has a countably infinite subset. Nonetheless, most of the "interesting" infinite sets have countably infinite subsets, even without Choice.)
Yeah, maybe the real messy problem is that we think of transfinite sets as having definite cardinality, or having cardinality at all, or even as being an actual set.
I suppose that's the problem, then. You tug at that thread and it unravels transfinite math.
It seems that cardinality is not just the ability to pair members of one set in a one-to-one correspondence with another set without having any in either set left over. If they cannot be paired in a way that exhausts one set but not another, then they don't seem to have the same cardinality. If that kills transfinite math, then so much the worse for transfinite math.
After all, shouldn't math be a truth-seeking endeavor, rather than a utility-seeking one?
Like most if not all mathematical concepts, "cardinality" is a stipulated concept.
Now, it may be that the concept does not map well onto the intuitive concept of "size". That's fine. But transfinite mathematics remains interesting, even if the concepts don't match.
Numerology remains interesting as well. it doesn't follow that numerology should be considered part of mathematics.
Are there interesting theorems of numerology?
The paper linked herein that the main results of the post are based on is forthcoming in Colloquium Mathematicum.
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