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

Compactness of the quantifier on "Complete Embedding of BA's"