[FOM] 327: Finite Independence/update

weierman@cage.ugent.be weierman at cage.ugent.be
Mon Jan 19 14:51:15 EST 2009

> 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


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
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.

Andreas Weiermann

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