[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)
joeshipman@aol.com
joeshipman at aol.com
Tue Aug 4 23:49:28 EDT 2009
Dunion's point is that CH results in a violation of a generalization of
Fubini's theorem (a set of points in R^2 that is countable on
horizontal lines and co-countable on vertical lines) which violates
reasonable intuitions.
The Banach-Tarski paradox violates intuition too, and the two
alternatives are to revise intuition or reject AC. Because AC is so
useful that it has become a standard axiom, most mathematicians have
not rejected it; instead they have revised their intuitions about the
possibility of assigning any subset of R^3 a "measure" that respects
finite additivity and rotational and translational invariance.
Freiling's "dart-throwing paradox" offers the same choice between
revising intuition or rejecting an axiom. But since CH is NOT a
generally accepted axiom, we should take intuition more seriously and
feel free to reject CH. Another way of putting this is that the
Freiling paradox is in one way even more counterintuitive than the
Banach-Tarski paradox -- because instead of saying that a suitable
measure can't be defined on arbitrary subsets of R^3, one is saying
that a suitable measure can't be defined on a much more restricted and
better-behaved class of subsets of R^2 (the ones which have measures
when restricted to horizontal and vertical subspaces, such that these
measures themselves cohere to form measurable functions allowing
"iterated integrals" to be defined).
One can imagine saying that some subsets of R^3 are just too wild to
have a reasonable measure, while maintaining that a subset of R^2 which
is countable on every horizontal or vertical line really ought to be
capable of being regarded as "small" in a suitable extension of
Lebesgue measure.
Dunion (and Freiling) have been misintrepreted. They are not claiming
to have an argument formalizable in ZFC; they are merely claiming that
mathematicians have overreacted to the results of Banach-Tarski, Godel,
and Cohen by throwing out too much of their intuition about assigning
measures to subsets of R^n. If there are going to be any generally
accepted additions to the standard ZFC axioms that settle CH (and if
you don't believe such additions are desirable then I'd like to know
why), they are going to have to have some kind of intuitive appeal, and
Freiling and Dunion are doing exactly what one would expect needs to be
done to make progress on this.
My personal view is that Freiling's argument proves too much. Any
subset of R^n of cardinality less than the continuum ought to be
regarded as "small" in my opinion, even if it is not
Lebesgue-measurable. Therefore I don't think that one should expect
extensions of Lebesgue measure to behave as nicely under products as
Lebesgue measure does. I would rather adopt the axiom that the
continuum is a real-valued measurable cardinal (which incidentally
implies the "strong Fubini theorem" Freiling wants to assume, even
though it doesn't support his dart-throwing argument).
We know that such a measure will not be invariant under translations
and rotations, but we already knew from Banach-Tarski that we have to
reject that part of our intuition about R^n (and we also know from
Einstein that space is not necessarily homogeneous and isotropic so
that doesn't damage the intuitive identification of space as a
continuum). Like Freiling and Dunion, I claim that we need not reject
all of our intuitions about the continuum just because we reject the
ones that conflict with ZFC.
-- JS
-----Original Message-----
From: Lasse Rempe <l.rempe at liverpool.ac.uk>
...It would be useful to provide some rationale
why the continuum having cardinality aleph_1 leads to more unusual
results than, say, the Banach-Tarski paradox. Furthermore, I would like
to know why you think these results should lead us to reject the
continuum hypothesis but not the axiom of choice.
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