Related papers: Naturality and Definability III

For a group $G$ first order definable in a structure $M$, we continue the study of the "definable topological dynamics" of $G$. The special case when all subsets of $G$ are definable in the given structure $M$ is simply the usual…

The primary goal of this paper is to establish a model of $ZFC$ wherein the definable tree property is affirmed for all uncountable regular cardinals. This endeavor commences with the utilization of both a supercompact cardinal and a…

In Machine Learning, an accepted definition of fairness of a decision taken by a classifier is that it should not depend on protected features, such as gender. Unfortunately, when constraints exist between features, such dependencies can be…

We classify the propositional modal validities arising from the category of sets under its natural classes of morphisms. The resulting validities depend on the morphism class, the size of the world, and the permitted substitution instances.…

The Svenonius theorem describes the (first-order) definability in a structure in terms of permutations preserving the relations of elementary extensions of the structure. In the present paper we prove a version of this theorem using…

We show that if we enrich first order logic by allowing quantification over isomorphisms between definable ordered fields the resulting logic, L(Q_{Of}), is fully compact. In this logic, we can give standard compactness proofs of various…

In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…

We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…

We consider natural cardinal invariants hm_n and prove several duality theorems, saying roughly: if I is a suitably definable ideal and provably cov(I)>=hm_n, then non(I) is provably small. The proofs integrate the determinacy theory,…

In this work, we propose the concept of Construction Defining Functionality (CDF), which characterizes functions by the structural spaces they generate through iteration,recursion, and logical application. By viewing functions as generators…

Recently, symbolic structures were proposed as finite representations of potentially infinite first-order structures, where Linear Integer Arithmetic terms and formulas define the domain and interpretations of a structure. We generalize…

Let $\mathcal M=(M,<,...)$ be a linearly ordered first-order structure and $T$ its complete theory. We investigate conditions for $T$ that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders…

Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) F{\o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of $\{0,1\}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = \{1,2,...,n\}$, the…

We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an…

We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself),…

We develop the notion of coherent ultrafilters (extenders without normality or well-foundedness). We then use definable coherent ultraproducts to characterize any extension of a model $M$ in any fragment of $\mathbb{L}_{\infty, \omega}$…

Deformation theory is treated for locally notherian formal schemes (non necessarily smooth). The cotangent complex is defined in the derived category through the homology localization functor. The basic properties and results of a…

A unification of characteristic mode decomposition for all method-of-moment formulations of field integral equations describing free-space scattering is derived. The work is based on an algebraic link between impedance and transition…

The work demonstrates that brain might reflect the external world causal relationships in the form of a logically consistent and prognostic model of reality, which shows up as consciousness. The paper analyses and solves the problem of…

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is…