[FOM] Absoluteness of arithmetic
Ali Enayat
ali.enayat at gmail.com
Tue Jan 2 14:00:41 EST 2007
In his message of Dec 31, 2006, Adam Kolany has asked whether it is possible
to have models M and M' of ZF which differ in their "arithmetical truths"
[in Kolany's notation: the (omega-)standard model N_M of arithmetic in the
sense of M is not elementarily equivalent to the (omega-) standard model
N_M' of arithmetic in the sense of M'].
The Godel-Rosser incompleteness theorem provides an immediate answer to the
question:
If ZF is consistent, then there is an arithmetical statement A such that the
theories ZF + A, and ZF + not A, are both consistent. So by the completeness
theorem of first order logic (also due to Godel) there are models M and M'
of the aforementioned extensions of ZF that are arithmetically incompatible.
It is also important to point out that by the MRDP theorem, the independent
sentence A can be chosen to be of a Diophantine nature. See Martin Davis'
excellent essay:
http://www.ams.org/notices/200604/fea-davis.pdf
for more in this direction [including a discussion of some of Harvey
Friedman's recent independence results].
Finally: a straightforward modification of the argument using Godel-Rosser
sentences yields the following result.
Theorem. Let T be a computably-enumerable consistent extension of ZF. There
are continuum-many models of T that are pairwise arithmetically
incompatible.
Ali Enayat
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