[FOM] 327: Finite Independence/update

[weierman@cage.ugent.be]

> I thought that it would be useful at this time to give a brief update
> on the state of the art regarding finite independence.
>
> NOTE: I did find
> http://www.maths.bris.ac.uk/~maaib//independence/node2.html
>   I don't know the author and I don't think it is up to date. Does
> anybody on FOM know if this is being maintained elsewhere?
>
Dear members of FOM,

As a complementary source for independence results
I would like to suggest the following survey of Andrey Bovykin
which is to appear in the proceedings of LC 2006

http://www.maths.bris.ac.uk/~maaib/new.pdf

The URL cited above by Harvey is also maintained by Andrey but
as Harvey noted an update might be welcome. (I think
the URL covers material until 2005.)


>
> Various aspects of these developments have been taken up by Bovykin
> (http://logic.pdmi.ras.ru/~andrey/research.html
> ), Carlucci (http://www.cis.udel.edu/~carlucci/research.html),
> Weiermann (http://wwwmath.uni-muenster.de/logik/Personen/weiermann/).

Let me add that I myself moved away from Muenster
via an intermediate stay in Utrecht to Ghent
(where I am currently building a research group
on the subject).
My current URL is therefore not the one mentioned above but
http://cage.ugent.be/~weierman/
This URL is slightly better updated then the one
from Muenster.

What I personally like very much about finite independence is
its connection to other fields like phase
transitions, logical limit laws, zeta universality, uniform distribution,
braid groups, analytic combinatorics and Tauberian theory.

>
> There has been work on extending the Hydra game to get independence at
> this level, and a bit higher. It does appear to be too complicated as
> it stands - although unifying and simplifying ideas may be possible:
>

In my point of view the Buchholz formulation of the Hydra battle
is very beautiful and I think it will be hard to make big simplifications
here. Buchholz's hydras come directly from ordinal notations and
simplifying the hydras would presumably require a simplified system
of notations. I would consider this as a very difficult challenge.

> W. Buchholz, An Independent Result for (Pi-1-1-CA + BI) + BI, Annals
> of Pure and Applied Logic, 33 (!987), 131-155.
>


Anyway, I admire Harvey's results very deeply and I am very interested
in the subject and its developments and so I am looking forward to
Harvey's next results on finite independence.

Best
Andreas Weiermann