[FOM] repairing a bridge between mainstream mathematics and f.o.m.

Arnon Avron aa at tau.ac.il
Sun Jul 26 06:07:33 EDT 2009

What will happen if our two heros through darts at the natrural numbers 
instead of the reals? It seems to me that by the same "argument"
we get that nothing could possoibly happen.

Arnon Avron

On Sun, Jul 12, 2009 at 11:13:40PM -0500, Tom Dunion wrote:
> Dear FOMers (and other readers),
> The purpose of this posting is two-fold: to heighten awareness of a conflict
> between a viewpoint prevalent within the f.o.m. community and another within
> ?mainstream? mathematics; and to suggest the direction of a possible resolution.
> To simplify things I will make the working assumption in this note that the
> continuum has cardinality aleph-two.
> First, a (very roughly stated) re-cap of the gist of the well-known argument of
> Chris Freiling against the CH, of ?throwing darts? at the unit interval.
> Toward a contradiction, assume CH and fix a bijection f, from the
> reals in [0,1] to the countable ordinals. Say x << y if f(x) < f(y)
> (as ordinals).  Two people stand in adjacent rooms and simultaneously
> throw idealized darts randomly at [0,1].  Wherever x lands, it has
> at most countably many predecessors, hence we have probability zero
> (i.e. Lebesgue measure zero) that we will have y << x; similarly,
> zero probability that x << y.
> But this thought experiment has an intuitive answer of a 50 per cent
> chance that x << y  (or that y << x), and there is nothing
> ?wrong? with such probabilistic intuition from
> the perspective of many mainstream mathematics practitioners.
> (As anyone who has taken actuarial exams can attest.)
> Also a well-known set theorist has found nothing disqualifying
> about this particular plausibility argument against the CH
> (beyond that if cannot be formalized in ZFC).
> Some have addressed Freiling?s argument in FOM, but not  persuasively,
> IMHO.  It is as if  there is a sign on the road between f.o.m. and basic
> measure theory, saying   ?Bridge out.  Go back.?  That does not bode well
> for a program of engaging mainstream mathematicians into a greater
> appreciation of foundations.
> My suggestion to try to resolve the conflict:  Suppose there exist
> nonmeasurable sets of cardinality aleph-one (recall, the working  assumption
> here is c = aleph-two). Contrary to the assumption of some people,
> under ZFC such a possibility (that not all less-than-continuum-sized
> sets must be measurable sets, with measure zero) is not automatically
> excluded from consideration.  If there is such a set A, a little measure
> theory gives that its outer measure is positive; also that other
> aleph-one-sized nonmeasurable sets exist, their outer measures taking
> values throughout (0,1].
> Finally, back to the dart throwing experiment.  Wherever y is, its set
> of ordinal predecessors {z: f(z) < f(y)} has cardinality not greater than
> aleph-one, but may well be a nonmeasurable set, not necessarily
> a set of measure zero; the same can be said about x.  So the
> ?contradiction? to our probabilistic intuitions that arose under CH does
> not arise here, and perhaps now we can take down the ?Bridge out? sign.
> I would welcome responses, pro or con, offlist or on, regarding this matter.
> Tom Dunion
> tom.dunion_at_gmail.com
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