Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Completeness in local positive logic

Arturo Rodriguez Fanlo, Ori Segel

TL;DR

This work builds the basic model theory of local positive logic, blending positive logic with local (ball-based) locality to study compactness, existential closure, and irreducibility in a multi-sorted setting. It establishes a Pi_1-local compactness theorem, develops locally positively closed models and their denials, and formulates a nuanced completeness theory featuring irreducibility, local joint continuation, and weak/completion notions. The paper also scrutinizes when locality is inherent, proving several equivalences in pointed languages and clarifying the separations among weak completion, completeness, and irreducibility. Together, these results lay foundational tools for studying locally definable structures and hyperdefinable phenomena within local positive logic, with planned extensions to types, saturation, and definability patterns in future work.

Abstract

We develop the basic model theory of local positive logic, a new logic that mixes positive logic (where negation is not allowed) and local logic (where models omit types of infinite distant pairs). We study several basic model theoretic notions such as compactness, positive closedness (existential closedness) and completeness (irreducibility).

Completeness in local positive logic

TL;DR

This work builds the basic model theory of local positive logic, blending positive logic with local (ball-based) locality to study compactness, existential closure, and irreducibility in a multi-sorted setting. It establishes a Pi_1-local compactness theorem, develops locally positively closed models and their denials, and formulates a nuanced completeness theory featuring irreducibility, local joint continuation, and weak/completion notions. The paper also scrutinizes when locality is inherent, proving several equivalences in pointed languages and clarifying the separations among weak completion, completeness, and irreducibility. Together, these results lay foundational tools for studying locally definable structures and hyperdefinable phenomena within local positive logic, with planned extensions to types, saturation, and definability patterns in future work.

Abstract

We develop the basic model theory of local positive logic, a new logic that mixes positive logic (where negation is not allowed) and local logic (where models omit types of infinite distant pairs). We study several basic model theoretic notions such as compactness, positive closedness (existential closedness) and completeness (irreducibility).
Paper Structure (4 sections, 26 theorems, 26 equations, 1 figure)

This paper contains 4 sections, 26 theorems, 26 equations, 1 figure.

Table of Contents

  1. Local languages and structures
  2. Local positive closedness
  3. Completeness
  4. Inherent locality

Key Result

Lemma 1.8

Let $\Gamma$ be a set of $\Pi_1$-local sentences. Then, $\Gamma$ is locally satisfiable if and only if $\Gamma\wedge {\mathtt{T}}_{\mathrm{loc}}$ is satisfiable. Furthermore, suppose $N\mathrel{\mathord{\models}_\star} \Gamma\wedge {\mathtt{T}}_{\mathrm{loc}}$ and pick $o\coloneqq (o_s)_{s\in\mathtt

Figures (1)

  • Figure 1:

Theorems & Definitions (84)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 1.8
  • proof
  • Theorem 1.9: Compactness theorem
  • ...and 74 more