Elementary Embedding

Definition

An elementary embedding is a function between two structures in model theory that preserves the truth of all first-order statements. This means that if a statement is true in one structure, it remains true when evaluated in the other structure after applying the embedding. Elementary embeddings play a critical role in understanding relationships between different mathematical structures, particularly in the context of completeness and equivalence.

5 Must Know Facts For Your Next Test

1. Elementary embeddings are crucial in model theory because they allow us to compare different models while preserving logical properties.
2. If an elementary embedding exists between two structures, it implies that they are elementarily equivalent, meaning they satisfy the same first-order properties.
3. The existence of an elementary embedding can provide insights into the consistency and completeness of different theories within mathematical logic.
4. Elementary embeddings are often used to demonstrate results like Löwenheim-Skolem theorem and completeness theorem, showcasing their importance in foundational mathematics.
5. A well-known example of elementary embeddings is the embedding of a countable model of set theory into a larger model, which preserves all first-order properties.

Review Questions

• How do elementary embeddings facilitate the comparison of different models in mathematical logic?
  • Elementary embeddings allow for a structured comparison between different models by ensuring that all first-order statements remain true when transitioning from one model to another. This preservation of truth enables logicians to draw meaningful conclusions about the properties of these models, leading to insights about their structure and relationships. For instance, if two models are related by an elementary embedding, they must share many important logical features.
• Discuss how elementary embeddings relate to the concepts of elementary equivalence and isomorphism in model theory.
  • Elementary embeddings establish a relationship between two structures by preserving truth values of first-order statements, which leads to the conclusion that these structures are elementarily equivalent. However, while all isomorphic structures are also elementarily equivalent, not all elementarily equivalent structures are isomorphic. This distinction is crucial because it highlights that elementary embeddings can exist even when the two models do not have a strict one-to-one correspondence in their elements or operations.
• Evaluate the implications of elementary embeddings on the completeness theorem and its significance in model theory.
  • The completeness theorem asserts that if a set of first-order sentences has a model, then there is a countable model for it. Elementary embeddings play a vital role in demonstrating this theorem by showing how one can embed models into larger ones while preserving first-order properties. This embedding leads to richer insights into the nature of mathematical theories and helps solidify our understanding of model existence. By connecting models through elementary embeddings, we gain tools to explore the boundaries and behaviors of various logical systems.

Related terms

Model Theory: A branch of mathematical logic that deals with the relationship between formal languages and their interpretations or models.

First-Order Logic: A formal logical system that allows for the expression of statements about objects and their relationships using quantifiers, variables, and predicates.

Isomorphism:

A type of mapping between two structures that shows a one-to-one correspondence, preserving the structure's operations and relations.