Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the downward Löwenheim-Skolem Theorem for elementary submodels

Matthias Kunik

Abstract

We introduce a new definition of a model for a formal mathematical system. The definition is based upon the substitution in the formal systems, which allows a purely algebraic approach to model theory. This is very suitable for applications due to a general syntax used in the formal systems. For our models we present a new proof of the downward Löwenheim-Skolem Theorem for elementary submodels.

On the downward Löwenheim-Skolem Theorem for elementary submodels

Abstract

We introduce a new definition of a model for a formal mathematical system. The definition is based upon the substitution in the formal systems, which allows a purely algebraic approach to model theory. This is very suitable for applications due to a general syntax used in the formal systems. For our models we present a new proof of the downward Löwenheim-Skolem Theorem for elementary submodels.
Paper Structure (5 sections, 9 theorems, 62 equations)

This paper contains 5 sections, 9 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Formal mathematical systems
  3. Models of formal mathematical systems
  4. The downward Löwenheim-Skolem Theorem
  5. Application to axiomatic set theory

Key Result

Lemma 3.3

Let $\mathcal{D}$ be a structure for $[M;\mathcal{L}]$. Then we have:

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • ...and 13 more