# [FOM] Why do set theorists dislike Chris Freiling's evidence against the continuum hypothesis?

Alasdair Urquhart urquhart at cs.toronto.edu
Thu Sep 2 13:49:06 EDT 2004

```Not being a set theorist, I don't know, but I'll hazard a guess.
Freiling's argument is quite closely related to another example
that was already considered by Goedel in his famous article
"What is Cantor's Continuum Hypothesis?"

It appears in the opening of Sierpinski's book on the continuum
hypothesis, and goes like this.  Consider the proposition:

There a partition of R^2 (the ordinary Euclidean plane) into
two sets A and B so that:

1.  Any horizontal line contains only countably many points of A.

2.  Any vertical line contains only countably many points of B.

Sierpinski proves this is equivalent to the continuum hypothesis.
Goedel discusses this and says it sounds implausible, but doesn't
consider it refutes the continuum hypothesis.

Now consider a "physical story" similar to Freiling's story to go with
Sierpinski's theorem.  I claim that using the continuum hypothesis,
I can colour the points of the Euclidean plane red or blue, so that
if you look along a horizontal line, all except a countable number
of points is blue, while if you look along a vertical line, all except
a countable number of points is red.  This sounds just like the
"object that is red and blue all over at the same time" that ordinary
language philosophers used to discuss when I was an undergraduate
philosophy student in Edinburgh.   The temptation is to say: "This is
impossible, the CH is false!"

But I agree with Arnon Avron.  We are importing our intuitions about ordinary
physical objects into a context where they make no sense.  The notion
of partitioning R^2 has nothing to do with colouring in the ordinary sense,
just as partitioning a ball (as in the Banach-Tarski "paradox") has nothing to
do with "cutting a ball into pieces" in the ordinary physical sense.
In the case of Freiling's argument, what sense does it make to
say "I threw a dart at the wall, and hit a point with rational coordinates."
None whatsoever!

Goedel himself has some good remarks on this kind of thing in Hao Wang's book
"From Mathematics to Philosophy."

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