# [FOM] Feferman's natural well-ordering problem.

Andreas Weiermann weiermann at math.uu.nl
Mon Feb 6 18:29:07 EST 2006

On Feb 1, 2006, at 10:33 PM, Bill Taylor wrote:
> This was mentioned by Andreas Weierman[n] the other day, and I can
> find no
> help on Googol for it.
> Can someone please explain what it is, for us?

I would like to make some additions. The well chosen reference by Chiari
contains a description of the current state of the art by a world leading
expert.

A reference by Feferman which mentiones the problem is:

http://math.stanford.edu/~feferman/papers/conceptualprobs.pdf

Another description is found in the second edition of
Takeuti's proof theory book in the appendix part by Feferman.

The point is that ordinal analysis is not just the calculation
of an ordinal of a reasonable mathematical theory
but to provide a natural well-ordering for the proof-theoretical
ordinal of the theory in question. For \epsilon_0,\Gamma_0,
the Bachmann-Howard ordinal, ...(and some more).. natural presentations
are well established.

Providing a convincing definition of a natural well-ordering
is still the challenge. (It is even not clear whether this
is a problem where a solution can be given.) My suggestion is to collect
natural properties of natural well-orderings to approximate the
problem.

Kind regards,
Andreas Weiermann