### Some compact logics --- results in ZFC

by Mekler and Shelah. [MkSh:375]

Annals Math, 1993

We show that if we enrich first order logic by allowing
quantification over isomorphisms between definable ordered fields
the resulting logic, L(Q_Of), is fully compact. In this
logic, we can give standard compactness proofs of various
results. Next, we attempt to get compactness results for some other
logics without recourse to diamondsuit, i.e., all our results are
in ZFC. We get the full result for the language where we quantify
over automorphisms (isomorphisms) of ordered fields in Theorem 6.4.
Unfortunately we are not able to show that the language with
quantification over automorphisms of Boolean algebras is compact,
but will have to settle for a close relative of that logic. This is
theorem 5.1. In section 4 we prove we can construct models in which
all relevant automorphism are somewhat definable: 4.1, 4.8 for BA,
4.13 for ordered fields. We also give a new proof of the
compactness of another logic -- the one which is obtained when a
quantifier Q_{Brch} is added to first order logic which says
that a level tree (definitions will be given later) has an infinite
branch. This logic was previously shown to be compact, but our proof
yields a somewhat stronger result and provides a nice illustration
of one of our methods.

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