[FOM] Kreisel Remark
kochen at math.princeton.edu
Tue Apr 1 08:47:51 EDT 2008
Here's the background on the Juliette Kennedy and Bob Solovay emails:
(i) The isomorphism of ultrapowers of real closed fields, using the
continuum hypothesis, was used in "Utraproducts in the Theory of
Models" to prove the completeness of their theory. I then remarked
that Kreisel could be invoked to get rid of CH for the completeness.
(ii)The isomorphism of ultraproducts of the p-adics and of power series
fields, using CH, was used in the paper "Diophantine Problems over Local
Fields I" with Ax to prove various arithmetic statements. Again, we
remarked that CH, via Kreisel and Godel, could be dispensed with for the
arithmetic consequences. However, we also remarked that it was possible
to eliminate CH directly by showing that our proof yielded isomorphic
elementary subfields of the two ultraproducts, giving the arithmetic
consequences. The same remark can be used in (i) to directly eliminate
CH from that proof. So the use of Kreisel is something of a red herring.
(iii) Shelah subsequently showed that it was impossible eliminate CH
from the proof of the isomorphism of the utraproducts in (ii).
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