Andreas Weiermann weiermann at math.uu.nl
Wed Jan 7 14:21:03 EST 2004

Dear members of FOM,

I also would like to make some remarks on FFF
and related stuff. In my point of view one
should interprete 
"mathematically natural independence results"
in terms of their intrinsic relation to pure and applied
mathematics. From that point of view H. Friedman's
results on the combinatorial well-foundedness
of ordinals and the combinatorial well-partial
orderedness lead, when one is interested
in threshold classifications, 
to rich and beautyful 
fields including analytic number theory, prime
number theory, tree enumeration,
combinatorial stochastics and the like.

Perhaps it is worth mentioning
that the profile process for ordinal terms
leads to Brownian excursions (following
Drmota and Gittenberger) and natural
parameters like the number of (different) summands
in a Cantor normal form shall follow (according
to some work in statistical mechanics) a Gaussian law
(when the context is chosen as expected).
I expect that such information can be used to describe in 
more detail the structure of descending sequences of ordinals.

When studying threshold functions for unprovability
it comes as a natural question whether there are
limit laws for ordinals. With Alan Woods the following
result has been obtained when the context is fixed
a la the random graph context. 
Fix alpha an additive principal 
less than epsilon nought. Let P be a first
order property formulated in the language of linear
orders. Pick beta less than alpha randomly.
Then P holds with probability zero or one.
If alpha is equal to epsilon nought then
a limit law still holds but not a zero one law.
(These results extend to larger prooftheoretic
ordinals and stronger logics. We expect Cesaro limit laws
for monadic second order logic.)

In my opinion such results on objects appearing
in foundations of mathematics are very useful
for advertising foundational research to

Best regards,
Andreas Weiermann

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