Complete theory

Definition

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.

5 Must Know Facts For Your Next Test

1. A complete theory ensures that any statement in the language can be definitively classified as either true or false within the context of that theory.
2. Morley's categoricity theorem states that if a complete theory is categorical in one infinite cardinality, it is categorical in all infinite cardinalities.
3. Complete theories can have many models; however, all models of a complete theory will satisfy the same first-order properties.
4. The completeness of a theory can impact how we interpret types and their relationships between different models.
5. Omitting types relates to complete theories by showing how certain types can be excluded from models while still preserving completeness.

Review Questions

• How does the concept of elementary equivalence relate to complete theories and their ability to categorize structures?
  • Elementary equivalence is essential for understanding complete theories since it establishes whether two structures can be considered identical in terms of their first-order properties. A complete theory guarantees that for every statement in the language, either it or its negation holds true, which means any two models of the same complete theory will satisfy the same sentences. Thus, if two structures are elementarily equivalent and belong to a complete theory, they share not only similar properties but also have corresponding behaviors within their respective interpretations.
• Discuss the implications of Morley's categoricity theorem on complete theories and their models across different cardinalities.
  • Morley's categoricity theorem highlights a significant property of complete theories: if such a theory is categorical in one infinite cardinality, then it must also be categorical in all infinite cardinalities. This means that all models of the complete theory at each infinite size will have similar structural characteristics. Therefore, this theorem reinforces the idea that complete theories can exhibit deep structural uniformity across different contexts, making them a powerful tool for understanding the relationships between various models.
• Evaluate how complete theories influence the interpretation of types and the applications of omitting types within model theory.
  • Complete theories have a profound impact on how we interpret types because they help define which properties can be consistently assigned to elements within models. When dealing with omitting types, we can strategically exclude certain types from models of a complete theory while ensuring that we still maintain the overall completeness of the theory. This interplay allows model theorists to create models tailored to specific requirements while exploring the boundaries of what is possible under a complete theoretical framework. Ultimately, this reveals how robust and flexible model theory can be when dealing with complex structures and their relationships.

Related terms

Elementary Equivalence: Two structures are elementarily equivalent if they satisfy the same first-order sentences, meaning there is no sentence in the language that distinguishes between them.

Categoricity: A theory is categorical in a particular cardinality if all models of that theory of that size are isomorphic, meaning they share a very specific structural similarity.

Types: A type is a set of formulas that describes the possible properties of an element in a model, playing a key role in understanding how theories can be manipulated and analyzed.