[FOM] Kreisel Remark

Simon Kochen 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).

Si Kochen

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