11.3 Complete theories and their properties
3 min read•july 30, 2024
Complete theories are the backbone of model theory, defining systems where every statement is either provable or disprovable. They're crucial for understanding formal systems' limits and capabilities, connecting to broader concepts like and decidability.
Diving into complete theories reveals their rich properties, from homogeneity to . This exploration leads us to stability theory, a powerful tool for classifying theories based on their models' behavior, with far-reaching applications in mathematics and logic.
Completeness of Theories
Defining Completeness
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- Completeness in model theory describes a theory where every sentence or its negation proves from the axioms
- means all sentences or their negations logically follow from the theory's axioms
- Theory of dense linear orders without endpoints exemplifies a complete theory
- of a given characteristic provides another example of completeness
- Incomplete theories contain sentences neither provable nor disprovable (theory of groups)
- Gödel's incompleteness theorems show consistent formal systems with arithmetic remain incomplete
- Completeness allows full characterization of a theory's models
Completeness vs Completeness Theorem
- Completeness of a theory differs from the completeness theorem in first-order logic
- Completeness theorem states a sentence proves if and only if true in all models
- Theory completeness focuses on provability within the specific theory
- Completeness theorem applies to the logical system as a whole
- Understanding the distinction clarifies the scope of completeness in different contexts
Completeness vs Decidability
Proving Equivalence
- Theory decides a sentence if it proves either the sentence or its negation
- "If" direction proof assumes theory T decides every sentence
- For any sentence φ in T's language, either T ⊢ φ or T ⊢ ¬φ, satisfying completeness definition
- "Only if" direction proof assumes T completes
- For any sentence φ, completeness implies T ⊢ φ or T ⊢ ¬φ, thus deciding every sentence
- Equivalence highlights completeness as theory's ability to prove or disprove all statements
- Proof relies on excluded middle principle in classical logic
Implications and Limitations
- Characterization of completeness crucial for understanding formal systems' limitations
- Gödel's incompleteness theorems demonstrate these limitations in arithmetic systems
- Decidability relates to algorithmic procedures for determining truth values
- Completeness does not guarantee decidability (some complete theories remain undecidable)
- Understanding the relationship between completeness and decidability informs theoretical computer science
Completeness and Categoricity
Defining Categoricity
- Categoricity describes a theory where all models of a given cardinality are isomorphic
- ℵ₀-categorical theory combines completeness and categoricity in some infinite cardinality
- implies complete theories with infinite models cannot be categorical in all infinite cardinalities
- Theory of dense linear orders without endpoints exemplifies both completeness and ℵ₀-categoricity
- characterizes ℵ₀-categorical theories using n-types over the empty set
Relationships and Applications
- Completeness does not imply categoricity (some complete theories have non-isomorphic models)
- Categoricity does not imply completeness (some categorical theories remain incomplete)
- Combination of completeness and categoricity provides powerful model-theoretic analysis tools
- Study of completeness-categoricity relationship advances stability theory
- Applications extend to classification theory in model theory
- Understanding these connections deepens insight into model-theoretic structures
Properties of Complete Theories
Homogeneity and Saturation
- Models of complete theories exhibit homogeneity and saturation
- Homogeneity allows extension of partial isomorphisms between finitely generated substructures to automorphisms
- Saturation means a model realizes all types over subsets with cardinality less than the model
- Monster model serves as a universal domain for studying complete theories
- Monster model exhibits high saturation and homogeneity
- Complete theories with model companion property have existentially closed models
- Omitting Types Theorem describes existence of certain models in complete theories
- Prime Model Theorem characterizes properties of specific models in complete theories
Stability Theory and Classifications
- Study of model properties in complete theories leads to stability theory development
- Stability theory classifies theories based on model behavior regarding saturation
- Classification extends to other model-theoretic properties (number of models, forking independence)
- Understanding stability aids in analyzing structure and behavior of models
- Applications of stability theory extend to algebraic geometry and number theory
- Classification results provide insights into model-theoretic complexity of theories
Key Terms to Review (18)
Categoricity: Categoricity refers to a property of a theory in model theory where all models of that theory of a certain infinite cardinality are isomorphic. This means that if a theory is categorical in a particular cardinality, any two models of that size will have the same structure, making them indistinguishable in terms of the properties described by the theory. This concept connects deeply with how theories and models behave under different axioms and the implications that arise from these relationships.
Compactness Theorem: The Compactness Theorem states that if every finite subset of a set of first-order sentences is satisfiable, then the entire set is satisfiable. This theorem highlights a fundamental relationship between syntax and semantics in first-order logic, allowing us to derive important results in model theory and its applications across mathematics.
Complete theory: A complete theory is a set of sentences in a formal language such that for any sentence, either that sentence or its negation is provable from the theory. This concept is crucial because it connects to the idea of elementary equivalence, which focuses on whether two structures satisfy the same first-order properties. A complete theory plays an important role in model theory, especially when discussing categoricity and how theories can be interpreted across different models.
Elementary embedding: An elementary embedding is a type of function between two structures in model theory that preserves the truth of all first-order formulas. This means if a property or relation holds in one structure, it holds in the other when corresponding elements are considered under the embedding, making it a crucial concept in understanding model relationships and properties.
Elementary Extension: An elementary extension is a model that extends another model in such a way that every first-order statement true in the original model remains true in the extended model. This concept is significant because it allows for the preservation of certain logical properties and structures while expanding the universe of discourse, making it a crucial idea in understanding various relationships between models.
Elementary model: An elementary model is a structure that satisfies all the axioms of a given theory in such a way that it is indistinguishable from any other model of that theory when viewed through the lens of first-order logic. This concept is crucial because it connects the ideas of model theory to the way we understand structures and their properties, particularly in relation to types and type spaces, as well as complete theories and their characteristics.
First-order completeness: First-order completeness refers to a property of a theory in which every statement that is logically implied by the axioms of the theory is also provable from those axioms. This concept is essential as it ensures that if something is true in all models of the theory, then it can be demonstrated through formal proof. This property connects deeply with the understanding of complete theories and their structures, and it plays a crucial role in the study of algebraically closed fields, where completeness can dictate the behavior and characteristics of these mathematical entities.
J. C. C. McKinsey: J. C. C. McKinsey was a prominent logician known for his contributions to model theory and the study of complete theories. His work focused on the properties of complete theories, specifically their categoricity and stability, influencing how we understand logical structures and their implications in mathematical frameworks.
Löwenheim-Skolem Theorem: The Löwenheim-Skolem Theorem states that if a first-order theory has an infinite model, then it has models of all infinite cardinalities. This theorem highlights important properties of first-order logic and models, demonstrating that certain structures can always be found, regardless of the size of the domain.
Morley's Categoricity Theorem: Morley's Categoricity Theorem states that if a complete first-order theory is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. This theorem highlights significant connections between model theory and set theory, showing how properties of theories can have far-reaching implications across different sizes of models.
Peano Arithmetic: Peano Arithmetic is a formal system that aims to capture the basic properties of natural numbers using axioms proposed by Giuseppe Peano in the late 19th century. It serves as a foundational framework for number theory, consisting of axioms that define the natural numbers and their operations, such as addition and multiplication. The structure of Peano Arithmetic lays the groundwork for understanding how mathematical statements can be formulated and the implications of consistency and completeness in formal theories.
Preservation of Formulas: Preservation of formulas refers to the property of certain logical theories whereby the truth of specific formulas remains intact when moving from one model to another. This concept is essential in understanding complete theories, which guarantee that every statement in the theory is either provably true or false, leading to a structured approach in deducing logical consequences and ensuring consistency across models.
Ryll-Nardzewski Theorem: The Ryll-Nardzewski Theorem is a fundamental result in model theory that characterizes complete theories through the existence of certain types of models. Specifically, it states that a complete theory is categorically equivalent to the existence of a model that is saturated and has a specific level of cardinality. This theorem connects to the properties of completeness and categoricity in model theory, emphasizing the behavior of models in relation to their theories.
Saturation: Saturation refers to a property of models in model theory where a model is considered saturated if it realizes all types that are consistent with its theory. This concept connects various features of model theory, including how models can be extended and the behavior of definable sets and functions within those models. Saturation plays a significant role in understanding the complexity and richness of models and their relationships to theories and types.
Second-Order Completeness: Second-order completeness refers to the property of a theory in which every second-order formula that is true in all models of the theory is provable from the axioms of that theory. This means that if a statement can be expressed in the language of second-order logic and holds in every model, then there exists a formal proof demonstrating its truth within that theory. This property emphasizes the distinction between first-order and second-order logics, showcasing the greater expressiveness of second-order logic and its implications for completeness.
Theory of Algebraically Closed Fields: The theory of algebraically closed fields is a collection of statements and principles that describe the properties of fields in which every non-constant polynomial has a root. These fields play a crucial role in algebra and model theory, particularly due to their completeness and the fact that they can be characterized by their algebraic closure, which guarantees solutions to polynomial equations. This leads to various implications regarding the completeness and categoricity of these theories.
Ultrapower: An ultrapower is a construction in model theory that allows us to create a new model by taking a product of structures and then factoring out an equivalence relation defined by an ultrafilter. This process reveals how properties of models can change when we consider their behaviors under the lens of infinite processes, linking the concept to important ideas like ultrafilters and the construction of ultraproducts.
Wilfrid Hodges: Wilfrid Hodges is a notable figure in model theory, recognized for his significant contributions to the understanding of logical structures, types, and model completeness. His work emphasizes the relationship between various logical frameworks and the properties of models, providing insights into the behavior of structures in mathematics and their applications in fields like algebraic geometry.