We show that for quasivarieties of p-algebras the properties of (i) having decidable first-order theory and (ii) having decidable first-order theory of the finite members, coincide. The only two quasivarieties with these properties are the trivial variety and the variety of Boolean algebras. This contrasts sharply, even for varieties, with the situation in Heyting algebras where decidable varieties do not coincide with finitely decidable ones.
This paper explores the coloring problem, focusing on the existence of uniformly colored substructures. The study primarily examines random graphs with edge coloring and their generic substructures. The key finding is that the absence of a monochromatic generic substructure corresponds to increased instability, meaning that the colored random graph hereditarily possesses the strict order property.
We classify all apartness relations definable in propositional logics extending intuitionistic logic using Heyting algebra semantics. We show that every Heyting algebra which contains a non-trivial apartness term satisfies the weak law of excluded middle, and every Heyting algebra which contains a tight apartness term is in fact a Boolean algebra. This answers a question of Rijke regarding the correct notion of apartness for propositions, and yields a short classification of apartness terms that can occur in a Heyting algebra. We also show that Martin-Löf Type Theory is not able to construct non-trivial apartness relations between propositions.
We study model theoretic characterizations of various collection schemes over from the viewpoint of Gaifman's splitting theorem. Among other things, we prove that for any and , the following are equivalent:
We study the meaning of “adding a constant to a language” for any doctrine, and “adding an axiom to a theory” for a primary doctrine, by showing how these are actually two instances of the same construction. We prove their universal properties, and how these constructions are compatible with additional structure on the doctrine. Existence of Kleisli object for comonads in the 2-category of indexed poset is proved in order to build these constructions.
Denoting by the free group over a two-element alphabet, we show in set-theory without the axiom of choice that the existence of a (2, 2)-paradoxical decomposition of free -sets follows from the conjunction of a weakened consequence of the Hahn-Banach axiom and a weakened consequence of the axiom of choice for pairs. The existence in of a paradoxical decomposition with 4 pieces of the sphere in the 3-dimensional euclidean space follows from the same two statements restricted to the set of real numbers. Our result is linked to the -paradoxical decompositions of free -sets previously obtained by Pawlikowski (, cf. [11]) and then by Sato and Shioya ( and , cf. [13]) with the sole Hahn-Banach axiom.
We prove that any definable family of subsets of a definable infinite set in an o-minimal structure has cardinality at most . We derive some consequences in terms of counting definable types and existence of definable topological spaces.
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