>"So the set of real numbers, while natural—indeed, immediately given— geometrically, nevertheless remains quite elusive: Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light."

Bah! Chaitin makes so much of his discovery that there is randomness in mathematics, but doesn't ever seem to have made the shift to thinking of structures as parametric over randomness.

To wit, generating random real numbers by simply flipping a countable number of coins and writing down the bits is trivial, and from that point of view:

1) An open set is just a set about which we can have finite information.

2) A closed set is just a set about which we can have complete information.

3) A compact set is just one we can quantify over and prove theorems about given only finite information.

4) A continuous function is one that has finite character: producing finite information about an element of the image only ever requires finite information about the argument.

These statements are a bit rough, but hey, if I ever have a gajillion dollars I can spend my newfound spare time merging the computational view on synthetic topology, Chaitin's algorithmic information theory, and the old "Age of Stochasticity" talk to get a view of mathematics that properly accounts for its finities and infinities.

http://www.stat.uchicago.edu/~lekheng/courses/191f09/mumford...