John McCarthy jmc at steam.stanford.edu
Wed Sep 1 15:45:14 EDT 2004

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

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.

Freiling also considers the consequences of a stronger axiom in which
"denumerable" is replaced by "of measure 0".

Godel believed the continuum hypothesis to be false and expressed hope
that someone would find intuitively acceptable axioms permitting
proving this fact.  He looked, however, for such axioms in set theory.

Nevertheless, I'm inclined to believe Godel would have liked
Freiling's axiom, coming from rather ordinary mathematics.

I'd like to understand why so many (almost all?) set theorists are so
sure that Freiling's axiom and the later axioms in his article
consider his work irrelevant to the continuum problem.

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