Eindhoven, November 4, 1994.
I have not been able to see that the presentation of Freiling's
axiom you gave (in your email of 27 Oct 1994 to Dahn) makes it
"intuitive". In my opinion the probability argument is just one of
those paradoxes we so easily get by careless formulation of
probability statements. A simpler form of the same "paradox" is the
following one.
For every natural number x the probability that an arbitrary
natural number y happens to be less than x is zero. Now throw darts
on the natural numbers to determine x and y. The probability that y
is less than x is zero, and the probability that x is less than y is
zero. Nevertheless, the probability that at least one of x<y and y<x
holds is equal to 1.
I think that in mathematics the notion "intuitive" is related to
a modification of a famous statement by E.Wigner about the undeserved
success of mathematics in the description of the real world. The
modification is about the undeserved success of incomplete or even
partially incorrect mathematics in describing at least parts of the
real world, and even parts of mathematics. In particular we try to
approach mathematical problems by taking the known situation in
some simple example as an indication of what happens in general,
often with success.
Handling such incomplete or defective mathematical knowledge
(consciously or sub-consciously) might be called "intuition".
If we get to know and understand more, we may get an intuitive
feeling that some of our previous intuitions are leading us in the
wrong direction. For example, I know intuitively that I have to be
extremely cautious, or even formal, when dealing with probability.
Your Exercise seems to require that one represents some intuitive
discussion in a formal system. This seems worth while trying. But it
might involve that we first have to put a mathematician on a sofa in a
psychiatrist's office in order to inspect the more subconscious parts
of intuive work.
N.G. de Bruijn.