Overview

Model Theory is the study of mathematical structures (like groups, fields, graphs) from the perspective of logic. It asks: What kind of mathematical structures satisfy a given set of axioms?

Core Idea

The core idea is semantics. While Proof Theory looks at the rules for manipulating symbols, Model Theory looks at what those symbols mean when interpreted in a mathematical structure.

Formal Definition

A model for a language is a structure $\mathcal{M}$ consisting of a domain (a set of elements) and interpretations for the symbols (constants, functions, relations). A sentence $\phi$ is true in $\mathcal{M}$ (written $\mathcal{M} \models \phi$) if the interpretation satisfies the condition.

Intuition

Think of a set of axioms as a blueprint (e.g., “It has three sides and three angles”). A model is a building that fits that blueprint (e.g., a specific triangle drawn on paper). Model theory studies the relationship between blueprints and buildings.

Examples

  • Non-Standard Analysis: Using model theory to create a model of the real numbers that includes infinitesimals (numbers smaller than any positive real number but greater than 0), justifying Leibniz’s calculus.
  • Independence Proofs: Proving that the Continuum Hypothesis is independent of ZFC by constructing models where it is true and models where it is false (Forcing).

Common Misconceptions

  • Misconception: A theory describes a single unique structure.
    • Correction: Most theories have many non-isomorphic models. For example, the axioms of Group Theory are satisfied by integers, matrices, symmetries, etc.
  • Misconception: It’s just about truth tables.
    • Correction: That’s propositional semantics. Model theory deals with infinite structures and complex quantifiers.
  • Proof Theory: The syntactic counterpart to model theory.
  • Isomorphism: A mapping showing that two structures are essentially the same.
  • Completeness Theorem: Gödel’s theorem stating that if a sentence is true in all models, it is provable.

Applications

  • Algebra: Used to solve problems in algebraic geometry and field theory (e.g., Ax-Kochen theorem).
  • Computer Science: Database theory and formal verification rely on model-theoretic concepts.

Criticism and Limitations

  • Abstractness: Can be extremely abstract, dealing with “monsters” (very large models) that have no physical reality.

Further Reading

  • Model Theory by C.C. Chang and H.J. Keisler
  • Model Theory: An Introduction by David Marker