[FOM] Choice of new axioms 1

Jacques Carette carette at mcmaster.ca
Mon Feb 13 20:00:15 EST 2006

Eray Ozkural wrote:

>I agree with Bauer in that the independence results hint at a
>fundamental relativism in mathematics rather than an
>absolute truth. 
What I find most fascinating here is the contrast with Computer 
Science:  the Church-Turing thesis has  proven itself to be quite 
``stable'', and so notions of computations can safely be determined via 
any Turing-complete programming language.  This gives rise to the very 
stable notion of Kolmogorov-Chaitin Complexity, which does not seem to 
have a satisfactory counterpart in mathematics (yet!).

This seems to be related to the strong countability assumptions of 
Turing machines, as well as some weaker finiteness assumptions (ie 
interesting programs terminate).  I am eagerly watching the developments 
in the ``realizability'' approach to Analysis and other parts of 
mathematics, as it appears that quite a bit of mathematics survives, and 
is sometimes reborn in quite fascinating new guises.

Has anyone tried to work on Kolmogorov type complexity theory for 
mathematical systems which are somehow ``stable''?


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