In this paper, the notions of countable S∗-compactness is introduced in L-topological spaces based on the notion of S∗-compactness. An S∗-compact L-set is countably S∗-compact. If L = [0, 1], then countable strong compactness implies countable S∗-compactness and countable S∗-compactness implies countable F-compactness, but each inverse is not true. The intersection of a countably S∗-compact L-s...

We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic defined by this Kripke frame on constant domains and vice versa. This allows us to transfer several recent results on Gödel logics to logics based on countabl...

Church’s Problem asks for the construction of a procedure which, given a logical specification φ on sequence pairs, realizes for any input sequence I an output sequenceO such that (I,O) satisfies φ. McNaughton reduced Church’s Problem to a problem about two-player ω-games. Büchi and Landweber gave a solution for Monadic Second-Order Logic of Order (MLO) specifications in terms of finite-state s...

We solve the question of the existence of a Poisson-Pinsker factor for conservative ergodic infinite measure preserving action of a countable amenable group by proving the following dichotomy: either it has totally positive Poisson entropy (and is of zero type), or it possesses a Poisson-Pinsker factor. If G is abelian and the entropy positive, the spectrum is absolutely continuous (Lebesgue co...

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X \ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of count...

The G0-dichotomy due to Kechris, Solecki and Todorčević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the G0-dichotomy for Σ 0 ξ-measurable countable colorings when ξ ≤ 3. A Σ 0 ξ-measurable countable coloring gives a covering of the diagonal consisting of countably many Σξ squares. This leads to the study of countable unions of Σξ recta...

Properties similar to countable fan-tightness are introduced and compared to countable tightness and countable fan-tightness. These properties are also investigated with respect to function spaces and certain classes of continuous mappings.

First Order ID-Logic interprets general first order, nonmonotone, inductive definability by generalizing the wellfounded semantics for logic programs. We show that, for general (thus perhaps infinite) structures, inference in First Order ID-Logic is completeΠ2 over the natural numbers. We also prove a Skolem Theorem for the logic: every consistent formula of First Order ID-Logic has a countable...

The set of automorphisms of a one-dimensional subshift (X,σ) forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy subshift. In particular, we show that the Baumslag-Solitar groups BS(1, ...

We introduce a necessary and sufficient condition for the ω-extensionality rule of higherorder equational logic to be conservative over first-order many-sorted equational logic for ground first-order equations. This gives a precise condition under which computation in the higher-order initial model by term rewriting is possible. The condition is then generalised to characterise a normal form fo...