  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 diamond, 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.
