Mathematics > Mathematical Logic > Model Theory

Model Theory:

Model Theory is a branch of mathematical logic that examines the relationships between formal languages, which are collections of symbols and rules used for constructing expressions, and their interpretations, or models, which provide meaning to these expressions. Essentially, it is the study of how mathematical statements can be understood in different structures.

Core Concepts:

1. Structures:
   • Signature: A signature \( \Sigma \) consists of a set of function symbols, relation symbols, and constant symbols, each with an assigned arity, which dictates how many arguments they take.
   • Model: Given a signature \( \Sigma \), a model \( M \) is a structure that assigns meaning to the symbols in \( \Sigma \). It specifies a domain \( D \) (a set of elements), interprets each function symbol as an actual function on \( D \), each relation symbol as a relation on \( D \), and each constant symbol as a specific element of \( D \).
2. Sentences and Formulas:
   • Language \( L_\Sigma \): A formal language built from a signature consisting of variables, logical connectives (such as \(\wedge\) for “and”, \(\vee\) for “or”, \(\neg\) for “not”), quantifiers (\(\forall\) for “for all”, \(\exists\) for “there exists”), and the symbols in \( \Sigma \).
   • Sentence: A formula with no free variables, which can be evaluated as either true or false in any given model.
   • Formula: An expression built from symbols in \( \Sigma \) that may contain free variables and which can be made into a sentence by specifying a domain for those variables.
3. Satisfaction and Validity:
   • Satisfaction: A model \( M \) satisfies a sentence \( \phi \), written \( M \models \phi \), if \( \phi \) is true in \( M \) when interpreted according to the domain and the meanings of the symbols as defined by \( M \).
   • Validity: A sentence \( \phi \) is valid, denoted \( \models \phi \), if it is true in every model of the appropriate signature.
4. Elementary Equivalence:
   • Two models \( M \) and \( N \) are elementarily equivalent, written \( M \equiv N \), if they satisfy the same sentences in a specific language \( L \).
5. Types and Realizations:
   • A type is a collection of formulas with free variables. If there exists some element in a model that satisfies all formulas in a type, the type is said to be realized in that model.

Applications and Importance:

Model Theory finds applications in various areas of mathematics, including algebra, topology, and combinatorics. For example, it helps in understanding the properties of algebraic structures like groups, rings, and fields by studying their models. Model Theory also plays a critical role in non-standard analysis and provides a framework for understanding the limitations and capabilities of formal systems.

Example:

Consider the language of groups, which includes a single binary operation symbol \(\cdot\) (interpreted as group multiplication), a constant symbol \(e\) (interpreted as the identity element), and a unary operation symbol \((\,)^{-1}\) (interpreted as taking inverses).

A typical group model \( G \) consists of a set \( D \) paired with functions:
- \( \cdot^G : D \times D \rightarrow D \) representing the group operation,
- \( e^G \in D \) representing the group identity element,
- \( (\,)^{-1}G : D \rightarrow D \) representing inversion.

In \( G \), the axioms of group theory (such as associativity \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)) must hold when the symbols \(\cdot\), \(e\), and \((\,)^{-1}\) are interpreted according to the model \( G \). The model \( G \) satisfies the group axioms if and only if \( G \models \text{Group Axioms} \).

Summary:

Model Theory provides the analytical tools required to study and interpret formal systems within the vast landscape of mathematical structures. By bridging symbolic logic and concrete mathematical entities, it deepens our understanding of both the theoretical foundations and practical implications of mathematical languages and their associated models.