John McCarthy (jmc@sail.Stanford.EDU)
Thu, 27 Oct 1994 02:44:28 -0700

It is too narrow a view of mathematics to consider it as only
concerning the consequences of an initially chosen set of axioms.
Consider Chris Freling's paper "Evidence Against the Continuum
Hypothesis", JSL 1982. As Goedel knew, Cohen (1965) had proved the
independence of the continuum hypothesis from the axioms of set
theory, complementing Goedel's 1940 result that the continuum
hypothesis was consistent with ZF. Nevertheless, 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.

Freiling did it only a few years after Goedel died. Freiling's ideas
was that people have intuitions not merely about numbers but also
about the real line, including the unit interval. Here is the
build-up to Freiling's axiom. Suppose you have a denumerably infinite
set A on the unit interval. Now you throw a dart at the unit interval
getting a number x. What is the probability that the x is in A?
0, you are supposed to answer. Fine says Freiling, let's elaborate
a little.

Let f be a function from the unit interval to the set of deunumeraable sets
in the unit interval, i.e. for each x, f(x) is a denumerable set of
points in the unit interval. Now throw a dart to determine x. Now
throw a second dart to to determine y..

What is the probability y is a member of f(x)? 0, you should say.
What is the probability x is a member of f(y)? 0, you should say.

Gotcha! say Freiling. We'll weaken what you have agreed to and
express it as axiom.

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).

>From this inuitive axiom the falsity of the continuum hypothesis
follows in two lines.

The coninuum hypothesis states that the cardinality of the unit
interval is the same as that of the set of denumerable ordinals.
Choose a 1-1 correspondence. Now for any x, let f(x) be the
set of smaller points in the ordering given by the correspondence.
If we now choose y, we msut have either y in f(x) or x in f(y),
according to whether y or x corresponds to a smaller ordinal.

I think Goedel would have been pleased with this result of Freiling's
and with Freiling's other results.

Determinined syntacticists, including unfortunately the editor of
Goedel's collected papers, find Freiling's work of no interest. I
suppose they don't like the idea of anyone having inuitions about the
unit interval worthy of being made into axioms. The whole discussion
given above should make a determined syntacticist nervous. He would
have to ask, "Precisely what symbol manipulations were you talking

Exercise: Put the above discussionion into PRA. See if another PRA
fan can figure out what the hell you are talking aobut from reading it.