  We try to build, provably in ZFC, for a first order T a model in which any
isomorphism between two Boolean algebras is definable. The problem, compared to
[Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do
not use Skolem function (which do not matter for proving compactness of logics
but still are of interest). Let lambda be 2^mu if regular and its successor
otherwise. Model theoretically we investigate notions of bigness of types,
usually those are ideals of the set of formulas in a model, definable in
appropriate sense. We build a model of cardinality lambda^plus by a sequence of
models M_alpha of cardinality lambda for alpha less than lambda^plus, each
M_alpha equips with a sequence (M_alpha, i, a_alpha, i, Omega_alpha, i) : i in
S_I subseteq lambda, with M_alpha, i is of cardinality less than lambda,
precedes-increasing continuous with i, Omega_alpha, i a bigness notion defined
using parameters from M_alpha, i and a_alpha, i realized in M_alpha, i plus 1
over M_alpha,i a Omega_alpha, i-big type. As alpha increase, not only M_alpha
increase, but this extra structure increasing modulo a club of lambda, this is
why we have insisted on lambda being regular. This can be considered as a way
to omit types of cardinality lambda, which in general is hard. The fact that
lambda is not too much larger than mu helps us to guarantee that any possible
automorphism of structures be defined in M equals union bracket M_alpha: alpha
less than lambda^plus bracket by approximations of cardinal mu and so we can
enumerate them all.

Logic