Theories

Definition

In model theory, a theory is a set of sentences in a formal language that is closed under logical consequence. This means that if a sentence can be derived from the sentences in the theory using logical rules, then that sentence is also included in the theory. Theories are foundational to understanding how models represent various structures and properties in mathematical logic.

5 Must Know Facts For Your Next Test

1. A theory can be seen as a collection of propositions that collectively describe some properties of a mathematical structure.
2. Different theories can be formed by varying the axioms, leading to distinct logical systems with their own interpretations.
3. A consistent theory is one where no contradictions can be derived, making it essential for it to have a model.
4. In model theory, one can talk about the completeness of a theory, which indicates whether all truths about a model can be derived from that theory.
5. The process of defining a theory often involves choosing axioms carefully to capture the desired properties of the structures being studied.

Review Questions

• How do theories relate to axioms and models in formal logic?
  • Theories are built upon axioms, which serve as foundational truths accepted without proof. Each theory consists of these axioms and additional sentences that follow logically from them. Models provide concrete examples that satisfy the sentences of a theory, showing how abstract concepts can be applied. Understanding this relationship helps in comprehending how logical frameworks are structured and analyzed.
• Discuss the implications of having a consistent versus an inconsistent theory in model theory.
  • A consistent theory is crucial because it means that no contradictions arise from its axioms and derived sentences, which ensures that at least one model exists that satisfies it. On the other hand, an inconsistent theory cannot have a model because contradictions imply that any statement can be derived, leading to an absence of meaningful interpretations. Thus, consistency is essential for maintaining the integrity and applicability of a theoretical framework in model theory.
• Evaluate how the completeness of a theory affects its utility in mathematical logic and model theory.
  • The completeness of a theory means that every statement that is true in all its models can be proven within the framework of the theory. This property enhances the utility of the theory since it provides assurance that all relevant truths can be derived, making it more robust and reliable for applications in mathematical logic. A complete theory facilitates understanding and exploration by ensuring that no truths about its models remain elusive, thereby empowering mathematicians and logicians to make comprehensive deductions and analyses.

Related terms

Axioms: Basic statements or propositions that are accepted as true without proof, serving as the starting point for further reasoning and arguments within a theory.

Models: Structures that satisfy the sentences of a theory, demonstrating how the abstract concepts within a theory can be realized in concrete terms.

Logical Consequence: A relation between sentences whereby one sentence logically follows from another or from a set of sentences, crucial for determining what belongs to a theory.