>> Axiom (Freiling): Let f be any function from the unit interval to
>> denumerable sets in the unit interval. Then there exist x and y
>> in the unit interval such that y is not in f(x) and x is not in f(y).
These ideas really go back to the 1920's. Sierpinski's book on the
Continuum Hypothesis (CH) (1934) contains many consequences of
(and equivalents to) CH; some of which seem a bit pathological.
In particular, it was known (I think, due to Sierpinski) that CH is
equivalent to the statement that the Euclidean plane can be covered by
countably many graphs of functions and inverse functions. This is
the same as saying that Freiling's Axiom is equivalent to (not CH).
As John McCarthy said,
>> Goedel believed that the continuum hypothesis was false and believed
>> that someone would come up with more intuitive acceptable axioms and
>> would then be able to prove it false.
Goedel undoubtedly was familiar with the work of Sierpinski and others,
and did not think of this as sufficient evidence. I think Goedel
would have seen right away that Freiling's Axiom adds nothing new.
As Ingo Dahn points out:
>> many natural subsets of the reals where shown to be either countable
>> or of power continuum (without use of CH). This was considered as an
>> argument in favour of CH.
This was also known by the 30's. This could be taken as an argument
that CH is irrelevant, since regardless of your set theory, CH is true
for any Borel set, which includes any set you're likely to construct
doing "ordinary" mathematical analysis.
Ken Kunen