There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ω CK 1 and where the LωCK 1 ,ω-theory of M is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ω-theory. Most well-known models with unbounded atoms below ωCK 1 also realize a non-principal LωCK 1 ,ω-type; such a model that preserves the Σ1-admissibility of ωCK 1 will hav...

The Scott process of a relational structure M is the sequence of sets of formulas given by the Scott analysis of M . We present axioms for the class of Scott processes of structures in a relational vocabulary τ , and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970’s, showing that a counterexample to Vaught’s Conjecture has models of cofinally many Scott ranks ...

We introduce a model of the second-order lambda calculus. Such a model is a Scott domain whose elements are themselves Scott domains, and in it polymorphic maps are interpreted by generic continous maps.

Let X be a dcpo and let L be a complete lattice. The family σL(X) of all Scott continuous mappings from X to L is a complete lattice under pointwise order, we call it the L-fuzzy Scott structure on X. Let E be a dcpo. A mapping g : σL(E) −> M is called an LM-fuzzy possibility valuation of E if it preserves arbitrary unions. Denote by πLM(E) the set of all LM-fuzzy possibility valuations of E. T...

In this paper we try to improve the current state of understanding concerning models of spaces with Scott domains. The main result given is that any developable space which has a model by a Scott domain must be Cech-complete. An important consequence is that any metric space homeomorphic to the maximal elements of a Scott domain must be completely metrizable.