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

[Tom Dunion]

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