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Affine logic with the integration operator

Seyed-Mohammad Bagheri

Abstract

Affine continuous logic is extended to affine integration logic. Affine compactness theorem is proved by both the ultramean construction and Henkin's method. Also, a proof system and a completeness theorem are given. An appropriate variant of the Keisler-Shelah isomorphism theorem holds in this setting. This helps us to characterize non-forking extensions in affine stable theories by means of the notion of elementary embedding in the expanded logic.

Affine logic with the integration operator

Abstract

Affine continuous logic is extended to affine integration logic. Affine compactness theorem is proved by both the ultramean construction and Henkin's method. Also, a proof system and a completeness theorem are given. An appropriate variant of the Keisler-Shelah isomorphism theorem holds in this setting. This helps us to characterize non-forking extensions in affine stable theories by means of the notion of elementary embedding in the expanded logic.
Paper Structure (6 sections, 47 theorems, 114 equations)

This paper contains 6 sections, 47 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Syntax and semantics
  3. Affine compactness
  4. Completeness
  5. Types
  6. The isomorphism theorem

Key Result

Theorem 2.3

(Kantorovich, Aliprantis-Inf Th. 8.32) Let $G$ be a majorizing vector subspace of a Riesz space $E$. Let $\Lambda:G\rightarrow\mathbb R$ be a positive linear function. Then $\Lambda$ has an extension to a positive linear function on $E$.

Theorems & Definitions (81)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • Lemma 2.8
  • proof
  • ...and 71 more