[FOM] A few observations on probability, cardinality, and the continuum

Tom Dunion tom.dunion at gmail.com
Fri Dec 24 00:44:13 EST 2010

The principle of Indifference tells us that if neither outcome of a
two-outcome experiment has any reason to be preferred, the probability
of each outcome is 50 per cent.  I would denote as the principle of
Coherence the idea that if an experiment (actual, or a thought
experiment) can be “naturally” described within the framework of two
different sample spaces, the probabilities of the various outcomes
should come out the same.

At the root of the difficulty of genuine paradoxes of probability that
are not mere mistakes in reasoning (such as the “Monty Hall Problem”)
we often find lack of Coherence: different but plausible sample spaces
yield different results.  (See, for example, “Sleeping Beauty”; also
J. Bertrand’s chords of a circle paradox, for simple illustrations of
this problem.)

With regard to the thought experiment of tossing 2 random darts at the
unit interval (Freiling’s Axiom of Symmetry), I submit that what has
so disquieted a lot of people is really a clash of these principles:
neither dart should be preferred to hit a point less than the other
(Indifference), but assuming the Continuum Hypothesis, a natural
description arises (by means of any well-ordering of [0,1] of length
omega_1) such that each dart is thrown at a countable set, with
probability zero of hitting the target, violating Coherence.

There may be a way out of the conundrum which has not been followed
up.  It may be that the cardinality of the continuum is (say) aleph_2,
yet the set of predecessors of each dart turns out to typically be a
nonmeasurable set. The existence of such sets of cardinality aleph_1
together with not-CH is known to be consistent with ZFC.

This leaves us with an argument against the CH which seems to lean
heavily on the principle of Indifference.  Can anything more be said,
even tentatively?

Well, one claim about nonmeasurable sets may be that they are
problematic, not because none of them can ever be assigned
intrinsically meaningful probabilities in any experiment, but rather,
they just don’t “play nice” with other sets.  Admittedly, to think
that *any* set of size aleph_1 has a probability of one-half of being
hit by a dart thrown into an interval whose cardinality is aleph_2 may
feel like an assault on our intuition;  but suppose (for about 30
seconds) the value of 2^{aleph_0} to be aleph_5.  Suppose further (and
this is still consistent with ZFC) that the smallest cardinality of a
set which is not of measure zero is aleph_4.  Does your intuition feel
as violated now?  If not, maybe it should not have felt so “wrong”
when thinking about those nonmeasurable sets of cardinality aleph_1.

Lastly, I don’t buy the claim that Freiling’s argument is essentially
self-refuting, i.e. throw 2 darts then 2^{aleph_0} cannot be aleph_1,
but throw 3 darts, then (by a clever use of mappings) it cannot be
aleph_2, etc.  That’s because one can “rig up” an argument using just
the first two darts thrown, if one appeals to further claims of
arguable plausibility.  I’ll just leave it there, since I don’t want
to go off on an excursus from the main point of all this: Indifference
plus Coherence seems to lead to a quandary that heavy-duty users of
mathematics (think: statisticians and physicists) can understand.  So
here is the “controversial” part of this posting, but meant as an
invigorating challenge, not as a put-down -– why should mainstream
mathematicians be expected to cheer on the larger f.o.m. community,
when that community cannot see its way clear to a resolution of such a
comprehensible (to mainstreamers) argument as Freiling’s?

Tom Dunion

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