[FOM] Naturalness

[Harvey Friedman]

On 1/27/06 9:44 AM, "Eray Ozkural" <examachine at gmail.com> wrote:

> On 1/27/06, Harvey Friedman <friedman at math.ohio-state.edu> wrote:
>> SO, there should be profound threshold phenomenon FOR ANY REASONABLE
>> LANGUAGE!!!!
>> 
>> Perhaps we should get to work on this...
> 
> Thanks for your fine post.
> 
> Your ideas sound quite similar to the approach of Chaitin and Calude
> using algorithmic information theory.
> 
There is a crucial difference. The threshold phenomena that I am talking
about begins at a much much earlier level of complexity than things coming
out of algorithmic information theory, which is closely tied to the axioms
of the theory in question.

The idea is that, say, in finite graph theory terms, independence from ZFC
comes much much earlier (as you go up in length) than it does in primitive
languages using standard encoding techniques of axioms from formal systems.

The reason one gets this kind of surprisingly early threshold is crucially
tied up with the exact nature of ZFC and the exact nature of, e.g., finite
graph theory concepts.

Harvey Friedman