Elementary embedding

Definition

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.

5 Must Know Facts For Your Next Test

1. Elementary embeddings allow for the transfer of properties between models, making them essential for understanding consistency and completeness of theories.
2. If there is an elementary embedding from model A to model B, then A can be viewed as a substructure of B that retains its original properties.
3. Elementary embeddings play a crucial role in proving the downward Löwenheim-Skolem theorem, showing that if a theory has an infinite model, it also has models of all infinite cardinalities.
4. Saturated models can be characterized using elementary embeddings, where every type realized in a model can be lifted to larger models through these embeddings.
5. Łoś's theorem utilizes elementary embeddings to provide a bridge between the concepts of first-order logic and the structure of models, ensuring that properties hold across various expansions.

Review Questions

• How do elementary embeddings relate to the concepts of consistency and completeness within theories?
  • Elementary embeddings are significant for understanding consistency and completeness because they preserve logical structure between models. If a theory is consistent, it has an elementary embedding into a larger model where the same logical truths apply. This means that if a theory can be shown to have an elementary embedding into another model, it implies that the properties defining its consistency and completeness are maintained across both structures.
• In what way does the downward Löwenheim-Skolem theorem utilize elementary embeddings to demonstrate model existence across different cardinalities?
  • The downward Löwenheim-Skolem theorem states that if a first-order theory has an infinite model, then it also has models of every smaller infinite cardinality. Elementary embeddings are key to this theorem as they show how you can take an infinite model and construct smaller models while preserving truth of all sentences. By identifying elementary embeddings from larger models to smaller ones, we can create models of various sizes that satisfy the same first-order sentences.
• Analyze how elementary embeddings contribute to the understanding of saturated and homogeneous models and their properties.
  • Elementary embeddings provide a framework for understanding saturated and homogeneous models by allowing us to explore how types can be lifted across different structures. In saturated models, any type that can be realized is reflected back into a larger context through these embeddings, ensuring that diverse properties are captured within homogeneous settings. This highlights how elementary embeddings not only preserve logical relations but also enable us to construct richer structures by revealing underlying connections among types and elements across various models.

Related terms

Isomorphism: A bijective mapping between two structures that preserves all relations and operations, establishing a strong form of equivalence.

Elementary equivalence: Two structures are elementarily equivalent if they satisfy the same first-order sentences, meaning any property expressible in first-order logic is true for both.

Model completeness: A property of a theory that ensures every embedding between models of the theory is an isomorphism, indicating a strong level of structural similarity.