We describe techniques for constructing models of size continuum inωsteps by simultaneously building a perfect set of enmeshed countable Henkin sets. Such models have perfect, asymptotically similar subsets. We survey applications involving Borel models, atomic models, two-cardinal transfers and models respecting various closure relations.

In [5], examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper, we give examples of complete ‐sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.

We prove that the existence of atomic models for countable atomic theories does not hold for Ln the first order logic restricted to n variables for finite n > 2. Our proof is algebraic, via polyadic algebras. We note that Lnhas been studied in recent times as a multi-modal logic with applications in computer science. 2000 MATHEMATICS SUBJECT CLASSIFICATION. 03C07, 03G15.

We produce a model of \ such that no outer model of \ has the same pure sets, answering a question asked privately by Eric Hall.