Model Theory

Model theory is a branch of mathematical logic that deals with the relations between formal languages and their interpretations or models. This field of teaching provides tools for analyzing the structure of mathematical theories and the types of models that make them true.

First, let's consider what a model is in this context. A model is a mathematical structure that gives meaning to the sentences of a formal language. For example, when we talk about a model of arithmetic, we mean a set of numbers with operations (such as addition and multiplication) and relations (such as greater-than or equality) that satisfy the axioms of arithmetic, such as Peano's axioms.

Formal languages and structures

In model theory, a formal language consists of symbols that can be put together to form sentences. These symbols typically include logical connectives such as ∧ (and), ∨ (or), ¬ (not), → (sign), as well as the quantifiers ∀ (for all) and ∃ (there exists). A language may also contain constant symbols, function symbols, and relation symbols.

The structure for a formal language includes a set called the universe, as well as interpretations for each of the function and relation symbols in the language. For example, the universe could be the set of natural numbers, and the interpretation of the sum symbol could be the usual summation operation on numbers.

For example, let's consider the following simple language:

• Static symbols: 0
• Single function symbol: S (successor)
• Binary relation symbol: =

The typical structure of this language includes:

• The set of natural numbers {0, 1, 2, 3, ...} is its universe.
• The constant 0 interpreted as the number 0.
• The function S is interpreted as the successor function (i.e., S(x) = x + 1).
• Relation = interpreted as numerical equality.

Truth and satisfaction

A sentence in this language might be something like ∀x (S(x) ≠ 0), which means "for every number x, its successor is not zero." This is a true statement in the structure of natural numbers because no natural number, when increased by one, is zero.

The concept of a model involves determining whether the sentences of a formal language are true (or satisfy) a condition within a particular structure. The model of a set of sentences is a structure in which all of those sentences are true.

Let's imagine the relationship between language and structure:

Example: group theory in model theory

As a practical example, consider group theory, a branch of abstract algebra that deals with algebraic structures known as groups. A group is a set equipped with an operation that satisfies certain axioms such as closure, associativity, identity, and inverse.

The language of group theory may include the following:

• Binary function symbol * (represents group operation)
• A constant symbol e (representing the identity element)
• The single function symbol inv (represents the inverse function)

Example of a sentence in this language: ∀x (x * e = x), which states that for all elements x, combining x with the identity element e via the group operation gives x.

A structure satisfying the axioms of group theory is a model of the theory of groups. For example, the integers under addition form a group.

    
Axioms:
- Closure: ∀x ∀y (x * y is defined).
- Associativity: ∀x ∀y ∀z ((x * y) * z = x * (y * z)).
- Identity: ∃e ∀x (x * e = x ∧ e * x = x).
- Inverse: ∀x ∃y (x * y = e ∧ y * x = e).

Model:
- Universe: Set of all integers {..., -2, -1, 0, 1, 2, ...}
- Operation: Addition
- Identity: 0
- Inverse: For x, the inverse is -x
    

Types of models

It is important to understand the different types of models to understand the reach and limitations of a theory. Some common classes of models are as follows:

• Finite models: Models where the universe is a finite set.
• Infinite models: Models that contain a universe of infinite sets, such as the set of all natural numbers.
• Standard models: Well-known models such as the normal structure for arithmetic.
• Non-standard models: Models that differ from standard models in unexpected ways. For example, non-standard analysis involves non-obvious entities such as infinitesimals.

Example: Finite and infinite models

Consider the theory of linear ordering proposed by ∀x ∀y ∀z ((x ≤ y ∧ y ≤ x) → x = y). This theory allows several models:

    
Finite Model:
- Universe: {1, 2, 3}
- Order: 1 ≤ 2 ≤ 3

Infinite Model:
- Universe: Natural numbers {0, 1, 2, ...}
- Order: Typical numerical order (0 ≤ 1 ≤ 2 ≤ ...)
    

The finite model has a limit of three elements, while the infinite model continues indefinitely.

Elementary embeddings and isomorphisms

An elementary embedding between two structures is a way to map one structure onto another, so that the truth of every sentence is preserved. When an elementary embedding is onto (surjective) and one-to-one (injective), it is called an isomorphism, indicating that the two structures are equivalent with respect to language.

Example of symmetry

Suppose we have two models of group theory:

    
Model A:
- Universe: {e, a, a²}
- Operation: e is the identity, a³ = e

Model B:
- Universe: {1, ω, ω²}
- Operation: 1 is the identity, ω³ = 1
    

These models are isomorphic under the mappings f(e) = 1, f(a) = ω, and f(a²) = ω² because they share a group structure (all follow the same rules).

Applications of model theory

Model theory is used in various areas of mathematics such as algebra, number theory, and combinatorics. Here are some areas where the principles of model theory are used:

• Algebra: Understanding the structure of various algebraic entities such as rings, fields, and vector spaces.
• Number theory: Understanding the properties of numbers through models that satisfy specific equations.
• Combinatorics: Describing finite structures that can often be extended to infinite settings.

Conclusion

Model theory bridges the gap between abstract logical frameworks and concrete mathematical structures. By defining what it means for a statement within a theory to be true in a given model, model theory enables the exploration and analysis of various mathematical systems.

Studying models provides deep insight into the logic, structure, and fundamental language of mathematics, which provides the basis for further mathematical exploration and innovation.

As the field continues to develop, model theory promises to reveal even more about the logical foundations of mathematics, thereby expanding our understanding not only of mathematical structures but also of the essence of mathematical truth.

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