[FOM] Interpretability in Q: more comments.

Robert M. Solovay solovay at Math.Berkeley.EDU
Fri Dec 24 04:45:31 EST 2004

I am going to have to mull over Harvey's letter before commenting on it.

Independently the question of hierarchy results occurred to me. Here's the
best I can do with my methods:

1) There is a true Pi^0_2 sentence A such that QQ[A] and QQ[not A] are
interpretable in Q. Here A is the sentence asserting that the following
function omega_2 is total:

	n --> n^{(log n) ^ (log log n)}.

This proposition A is sometimes known as Omega_2. If we dropped Omega_1
from QQ [getting a subtheory of IDelta_0 which I will call QQ-] then  we
could use Omega_1 for A: Both QQ-[Omega_1] and QQ-[not Omega_1] are
interpretable in Q.

2) Again if we drop the Omega_1 from the formulation of the A and B of my
last letter they become Sigma^0_2.

3) Most [perhaps all] of Nelson's interpretations just use the original +
and * restricted to a subdomain. For these interpretations [call them
internal] compatibility is clear. Then the interesting question is to
characterize the theory of all sentences which can be "internally
interpretable" in Q. A reasonable conjecture is that this theory is
generated by IDelta_0 + {Omega_n}_n. Perhaps this is easy to show or to
refute. I haven't yet given the matter much thought.

--Bob Solovay

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