[FOM] First order axiomatizability

François Dorais fgdorais at gmail.com
Thu Nov 24 23:24:52 EST 2011

Keisler's thesis "Ultraproducts and elementary equivalence" (UC
Berkeley, 1961) contains many characterizations of this sort. Most of
Keisler's results assumed GCH, but that hypothesis was later
eliminated by Shelah ("Every two elementarily equivalent models have
isomorphic ultrapowers," Israel J. Math. 10, 1971, 224–233). Keisler's
results can also be found in Bell & Solomson "Models and
Ultraproducts" (North-Holland, 1969; reprinted by Dover).

I think your proposed characterization should be closed under
ultraproducts instead of just ultrapowers. For example, the class of
all fields of nonzero characteristic is closed under isomorphisms and
ultrapowers, but it is not first-order axiomatizable. You may need an
additional hypothesis depending on exactly what you mean by
first-order axiomatizable. For example, the class of uncountable
fields is closed under isomorphisms and ultraproducts, but there are
plenty of countable fields. Keisler showed that a class of structures
that is closed under elementary equivalence and ultraproducts is
indeed the class of all models of some first-order theory.

François G. Dorais

On Thu, Nov 24, 2011 at 2:40 PM,  <pax0 at seznam.cz> wrote:
> Does someone know where there is a proof (and whose result it is) the following:
> A class of first order strucutres is fisrt order axiomatizable
> iff
> it is closed under isomorphisms and ultrapowers.
> Thank you Jan Pax
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