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

[joeshipman@aol.com]

The problem with Freiling's argument is that it proves too much. It 
depends on an intuition that any subset of the unit square or interval 
has an associated probability of being hit by a "dart".

If this probability is to be countably additive, you are assuming a 
real-valued measurable cardinal, which already disproves CH very 
strongly without any need for Freiling's symmetry argument.

If it is only finitely additive, then such measures exist (and in 1 or 
2 dimensions they are even compatible with Lebesgue measure and 
invariant under rigid motions), but you can't really be sure countable 
sets ought to have probability 0 of being hit.

If it is countably additive on points but not subsets, then you can ask 
why shouldn't it be k-additive on points for any cardinal k smaller 
than the continuum, in which case Freiling's symmetry axiom is 
inconsistent.

I say it's better just to go with the intuition that every subset of 
space has a "mass" which leads to a real-valued measurable cardinal. By 
Banach-Tarski, such a measure can't be invariant under rotations, which 
means that either our intuition about the isotropy of space has to be 
thrown out or our intuition about the continuity and infinite 
divisibility of space has to be thrown out.

There is an interesting alternative history here which has some useful 
foundational lessons. Because Banach-Tarski was discovered after the 
intuition of infinitely divisible space had been called into question 
by the atomic theory, it was interpreted as evidence against infinite 
divisibility rather than evidence against isotropy; however, if physics 
had developed differently (for example if Riemann had lived long enough 
to understand Maxwell's work and developed special and general 
relativity before 1900) then it is possible that the intuition of 
isotropy would have been discarded first, in which case  Banach-Tarski 
would not have been interpreted as evidence against infinite 
divisibility and the RVM axiom would have seemed much more intuitively 
plausible.  (Of course the intuition of infinite divisibility would 
eventually have been discarded too, but in the meantime RVM might have 
become the basis for an alternative axiomatization of set theory, a 
stronger one which proved among other things Con(ZFC), and CH might 
have been considered unproblematically false.)

I don't believe that any alien mathematical civilization would ever 
disagree with us about statements of arithmetic (they might think they 
had proven X when we didn't, or think that we hadn't proven Y when we 
claimed to, but there would never be a case where they thought they had 
proven X and we thought we had proven not-X, if X was arithmetical and 
the proofs had been sufficiently checked for mistakes). But they could 
easily disagree with us about things like CH.

-- JS


-----Original Message-----
From: Tom Dunion <tom.dunion at gmail.com>
To: fom at cs.nyu.edu
Sent: Fri, Dec 24, 2010 12:44 am
Subject: [FOM] A few observations on probability, cardinality, and the 
continuum


The principle of Indifference tells us that if neither outcome of 
atwo-outcome experiment has any reason to be preferred, the 
probabilityof each outcome is 50 per cent.  I would denote as the 
principle ofCoherence the idea that if an experiment (actual, or a 
thoughtexperiment) can be “naturally” described within the framework of 
twodifferent sample spaces, the probabilities of the various 
outcomesshould come out the same.At the root of the difficulty of 
genuine paradoxes of probability thatare not mere mistakes in reasoning 
(such as the “Monty Hall Problem”)we often find lack of Coherence: 
different but plausible sample spacesyield different results.  (See, 
for example, “Sleeping Beauty”; alsoJ. Bertrand’s chords of a circle 
paradox, for simple illustrations ofthis problem.)With regard to the 
thought experiment of tossing 2 random darts at theunit interval 
(Freiling’s Axiom of Symmetry), I submit that what hasso 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 naturaldescription arises (by 
means of any well-ordering of [0,1] of lengthomega_1) such that each 
dart is thrown at a countable set, withprobability zero of hitting the 
target, violating Coherence.There may be a way out of the conundrum 
which has not been followedup.  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 anonmeasurable set. The existence of such 
sets of cardinality aleph_1together with not-CH is known to be 
consistent with ZFC.This leaves us with an argument against the CH 
which seems to leanheavily on the principle of Indifference.  Can 
anything more be said,even tentatively?Well, one claim about 
nonmeasurable sets may be that they areproblematic, not because none of 
them can ever be assignedintrinsically meaningful probabilities in any 
experiment, but rather,they just don’t “play nice” with other sets.  
Admittedly, to thinkthat *any* set of size aleph_1 has a probability of 
one-half of beinghit by a dart thrown into an interval whose 
cardinality is aleph_2 mayfeel like an assault on our intuition;  but 
suppose (for about 30seconds) the value of 2^{aleph_0} to be aleph_5.  
Suppose further (andthis is still consistent with ZFC) that the 
smallest cardinality of aset which is not of measure zero is aleph_4.  
Does your intuition feelas 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 essentiallyself-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 bealeph_2, etc.  That’s because one can “rig up” 
an argument using justthe first two darts thrown, if one appeals to 
further claims ofarguable plausibility.  I’ll just leave it there, 
since I don’t wantto go off on an excursus from the main point of all 
this: Indifferenceplus Coherence seems to lead to a quandary that 
heavy-duty users ofmathematics (think: statisticians and physicists) 
can understand.  Sohere is the “controversial” part of this posting, 
but meant as aninvigorating challenge, not as a put-down -– why should 
mainstreammathematicians be expected to cheer on the larger f.o.m. 
community,when that community cannot see its way clear to a resolution 
of such acomprehensible (to mainstreamers) argument as Freiling’s?Tom 
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