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

TL;DR

This paper proves in ZFC the existence of models where all Boolean algebra isomorphisms are definable, using a novel approach involving bigness notions and type omission techniques, avoiding Skolem functions.

Contribution

It introduces a new method to build models with definable isomorphisms of Boolean algebras, extending prior work by handling pseudo-finite cases without Skolem functions.

Findings

Models with all Boolean algebra isomorphisms are definable in ZFC.

A new construction method using bigness notions and type omission.

Automorphisms are approximated by increasing sequences of models.

Abstract

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.