Omitting types in fragments and extensions of first order logic

Fix \(2 < n < \omega\). Let \(L_n\) denote first order logic restricted to the first $n$ variables. Using the machinery of algebraic logic, positive and negative results on omitting types are obtained for \(L_n\) and for infinitary variants and extensions of \(L_{\omega, \omega}\).

Atom canonicity and first order definability in classes of algebras of relations

Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever 𝔄 is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly clasfsical. It is also shown that any class K such that , where CRCAn is the class of completely representable CAns, and Sc denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that is not elementary, where Sd denotes the operation of forming dense subalgebra.

On the multi-dimensional modal logic of substitutions

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

Omitting types in logic of metric structures

This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete theory if and only if it is not principal, this is not true for the incomplete types by a result of Ben Yaacov. We prove that there is no simple test for determining whether a type is omissible in a model of a theory [Formula: see text] in a countable language. More precisely, we find a theory in a countable language such that the set of types omissible in some of its models is a complete [Formula: see text] set and a complete theory in a countable language such that the set of types omissible in some of its models is a complete [Formula: see text] set. Two more unexpected examples are given: (i) a complete theory [Formula: see text] and a countable set of types such that each of its finite sets is jointly omissible in a model of [Formula: see text], but the whole set is not and (ii) a complete theory and two types that are separately omissible, but not jointly omissible, in its models.