Related papers: Applicable Mathematics in a Minimal Computational …
Much like admissibility is the key concept underlying preferred semantics, strong admissibility is the key concept underlying grounded semantics, as membership of a strongly admissible set is sufficient to show membership of the grounded…
This the first of a series of articles dealing with abstract classification theory. The apparatus to assign systems of cardinal invariants to models of a first order theory (or determine its impossibility) is developed in [Sh:a]. It is…
We present a method for constructing countable models of small theories and apply it to prove theorems on the maximal number of countable non-isomorphic models of linearly ordered theories.
Reachability analysis, in general, is a fundamental method that supports formally-correct synthesis, robust model predictive control, set-based observers, fault detection, invariant computation, and conformance checking, to name but a few.…
Motivated by team semantics and existential second-order logic, we develop a model-theoretic framework for studying second-order objects such as sets and relations. We introduce a notion of abstract elementary team categories that…
This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on Realisability and the other on Sheaf Models in Algebraic Set Theory.
This work is motivated by the problem of finding the limit of the applicability of the first incompleteness theorem ($\sf G1$). A natural question is: can we find a minimal theory for which $\sf G1$ holds? We examine the Turing degree…
When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to…
Rice's theorem shows that nontrivial extensional properties of partial recursive functions are undecidable. For finite weighted Boolean optimization/CSP-style slices, a Rice-style structural analogue holds for tractability classification:…
Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $\beta$-models:…
We introduce a set of eight universal Rules of Inference by which computer programs with known properties (axioms) are transformed into new programs with known properties (theorems). Axioms are presented to formalize a segment of Number…
This paper describes a type of infinitary computer (a hypercomputer) capable of computing truth in initial levels of the set theoretic universe, V. The proper class of such hypercomputers is called a universal hypercomputer. There are two…
We explore a kind of first-order predicate logic with intended semantics in the reals. Compared to other approaches in the literature, we work predominantly in the multiplicative reals $[0,\infty]$, showing they support three generations of…
We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of…
We survey the logical structure of constructive set theories and point towards directions for future research. Moreover, we analyse the consequences of being extensible for the logical structure of a given constructive set theory. We…
Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange…
We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory $T$ is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model $M$ of $T$ has an expansion…
Similar to a tree grammar, a Horn theory can be used to describe an infinite set of terms. In this paper, we present a class of Horn theories such that the set of definable predicates is closed wrt. conjunction and such that the…
This paper deals with formulas of set theory which force the infinity. For such formulas, we provide a technique to infer satisfiability from a finite assignment.
In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain…