Definable families of theories, related calculi and ranks

Abstract

We consider sentence-definable and diagram-definable subfamilies of given families of theories, calculi for these subfamilies, as well dynamics and characteristics of these subfamilies with respect to rank and degree.

Formulas and properties

“…Definition [4]. If T is a family of theories and Φ is a set of sentences, then we put T Φ = ϕ∈Φ T ϕ and the set T Φ is called (type-) or (diagram-)definable (by the set Φ) with respect to T , or (diagram-)T -definable, or simply d-definable.…”

“…Proposition 1.9 [4]. For any ordinals α ≤ β, if RS(T ) = β then RS(T ϕ ) = α for some (α-ranking) sentence ϕ.…”

Algebras for definable families of theories

“…Throughout the paper we consider families T of complete first-order theories of a relational language Σ = Σ(T ) and use the following terminology in [1,2,3,4,5,6,7,8,9,10].…”

“…Definition [1]. If T is a family of theories and Φ is a set of sentences, then we put T Φ = ∩ As noticed in [1]…”

“…We define algebras associated with sentence-definable and diagram-definable subfamilies [1] as well as related characteristics connected with given families of theories. Topological properties and ranks for these algebras are characterized.…”