On completions, neat embeddings and omittings types, yet again

TL;DR

This paper explores deep algebraic logic results to prove new omitting types theorems for finite variable fragments of first order logic, revealing limitations and connections in algebraic and logical frameworks.

Contribution

It introduces new omitting types theorems for L_n, demonstrates failures under clique guarded semantics, and explores connections between strong and complete representability using rainbow and Monk-like constructions.

Findings

Proves that theories with quantifier elimination omit fewer types than possible.

Shows failure of omitting types theorem for L_n under clique guarded semantics.

Constructs examples illustrating the limits of representability and omitting types in algebraic logic.

Abstract

In this paper we investigate using the methodology of algebraic logic, deep algebraic results to prove three new omitting types theorems for finite variable fragments of first order logic. As a sample, we show that it T is an L_n theory and |T|=lambda, lambda a regular cardinal, if T admits elimination of quantifiers, then T omits < 2^{\lambda} many non isolated {\it maximal} types. This is basically a result of Shelah's restricted to L_n. that is not completely representable. We also show, using a rainbow construction for cylindric algebras, that the omitting types theorem fails for L_n even if we consider clique guarded semantics. This is done by constructing a an atomic \A\in \PEA_n with countably many atoms (which are coloured graphs) who Sc (Pinter's) reduct is not in S_c\Nr_n\Sc_{n+3}, but $A$ is elementary equivalent to a countable completely representable (polyadic equality) algebra. Various connections between the notions of strong representability and complete representability are given in terms of neat embeddings. Several examples, using rainbow constructions and Monk-like algebras are also given to show that our results are best possible. As a sample we show that, assuming the existence of certain finite relation algebras, that for any k\in \omega, there exists \A\in {\sf RPEA}_n\cap \Nr_n\PEA_{n+k} such that Rd_{\sf Sc}\Cm\At\A\notin S\Nr_n\Sc_{n+k+1}. This implies that for any finite n\geq 3, for any k\geq 0, there is an L_n theory and a type \Gamma such that Gamma is realized in every n+k+1 relativized smooth model, but cannot be isolated by a witness using n+k variables.