Good afternoon AIT mailing list,
I wrote imprecisely when I said that there are finitely many propositions that
are both true and provable. What I meant was that the consistency of mathematics
built upon a finite number of axioms depends on excluding Gödel numbers that have
a Kolmogorov Complexity beyond a particular threshold. This perspective is partly
motivated by Chaitin's heuristic argument:
If a set of theorems constitutes t bits of information,
and a set of axioms contains less than
t bits of information, then it is
impossible to deduce these theorems from these axioms.
-Chaitin, Information-theoretic Limitations on Formal Systems(1974)
In Chaitin's lectures 'From Philosophy to Program Size' he argues that this will
force future mathematicians to consider tradeoffs between adding new axioms
or accepting experimental methods as methods of verification in pure mathematics.
I believe that we have reached this point in the history of mathematics.
I won't argue that I have done what is necessary to prove a rigorous variant of
Chaitin's heuristic argument. But, I wonder whether there is a community of
information theorists that has made a serious effort in this direction. The best
approximation to what I am looking for appears to be:
Is Complexity a Source of Incompleteness? by Calude and Jürgensen(2004)