Moses Klein klein at math.ohiou.edu
Thu Sep 2 17:15:12 EDT 2004

John McCarthy said:
> I don't understand why almost all set theorists ignore, i.e. don't
> refer to Chris Freiling's paper offering evidence for the negation of
> the continuum hypothesis.  It seems to me that Gödel would have
> considered it the kind of intuitive axiom he wanted.
> Freiling's axiom is that if f maps [0,] into denumerable subsets of
> [0,1], then there exist a and b in [0,1] such that b is not in f(a),
> and a is not in f(b).
> Given ZF, the axiom is equivalent to the negation of the continuum
> hypothesis.

This statement, like the closely related one Alasdair Urquhardt
posted, is provably true with [0,1] replaced by a set of cardinality
aleph-1, and provably false with [0,1] replace by any set of higher

This is why I do not find it a convincing argument against CH. Any
counterintuitive quality it has derives from the sense that countable
subsets of the real line are "almost empty". But the same intuition
which would justify that, to me, would also justify considering
countable sets to be "almost empty" subsets of *any* countable set.
Since the analog of Freiling's axiom for omega_1 is provable in ZFC, I
am committed to it. There is no leap in credibility, then, to suppose
that the real line might have the same property.

> Freiling's intuitive argument for the axiom is based on probability
> theory and, I suppose, elementary measure theory.  He supposes that a
> is determined by throwing a dart at the unit interval and b by
> throwing a second dart.  Since f(a) is denumerable, the probability
> that b will land in in f(a) is zero.  Likewise, the probability that a
> will be in f(b) is zero.  Therefore, the probability neither b is in
> f(a) nor a in f(b) is 1.

Note that no measure theory is involved here beyond the fact that
countable sets have measure zero. Measure theory is just a way to get
at a natural sigma-ideal which includes all countable sets. But we
already have a natural sigma-ideal on any uncountable set: the ideal
of countable subsets itself.

Nonmeasurable sets are involved here: assuming CH, if f is a
counterexample to Freiling's axiom, then {(a,b): b in f(a)} is
nonmeasurable. We should be used to probability-theoretic notions not
being coherently applicable to nonmeasurable sets. So I don't find
this line of reasoning any more convincing.

Moses Klein
Visiting Assistant Professor
Department of Mathematics
Ohio University
Athens, OH  45701,  USA
klein at math.ohiou.edu

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