FOM: perfect independence results

Andreas Weiermann weierma at
Wed Jan 9 10:51:27 EST 2002

Dear members of FOM,

some time ago there has been a discussion in
FOM about perfect independence results which 
I found quite interesting and I have
arrived at some results which might be
of some interest with respect to this.
As pointed out in the discussion some people
might think that the Paris Harrington result (PHT) 
is not perfect in the sense that
it is based on a largeness condition.

To me it seems that PHT is not that artificial
for the following reason.
Let PH_f be the modified statement where
the largeness condition is replaced
by card(X) >= f(min(X)).

By Erdoes Rado 1952 we have that PH_f is 
provable in PA in case that f is log*
the inverse of the superexponential
function. Thus largeness is naturally
associated with PA for log*.
If f grows slightly faster than
log* than PH_f is independent.
In fact if f=id than we arrive
at PHT but even if f is a fixed
iterate of the binary length function
than PH_f is (as far as I can see) independent of PA.
In conclusion PH_f is independent for
all f which grow slightly faster than bounding
functions resulting from the Erdoes
Rado analysis of the finite Ramsey theorem.

Best regards,
Andreas Weiermann

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