Ultrafilters and ultrapowers are powerful tools in model theory. They allow us to extend models while preserving their properties, providing a way to construct new mathematical structures with desired characteristics.

Ultraproducts, built using ultrafilters, play a crucial role in proving fundamental theorems in logic. They're used in the compactness and completeness theorems, and have applications in various mathematical fields, from non-standard analysis to algebraic geometry.

Ultrafilters and Ultrapowers

Ultrafilters and ultrapowers in model theory

• Ultrafilters maximize inclusion on a set I filter by containing either A or its complement for any subset A of I
• Ultrafilter intersections of finite sets remain non-empty
• Principal ultrafilters generated by single element while non-principal not generated by any single element
• Ultrapowers extend model M elementarily by constructing $M^I$ and defining equivalence relation with ultrafilter on I
• Ultrapowers preserve all first-order properties of original model

Construction of ultraproducts

• Select model family $(M_i)_{i \in I}$ indexed by I and ultrafilter U on I
• Form direct product $\prod_{i \in I} M_i$ and define equivalence relation $\sim_U$
• Ultraproduct emerges as quotient set $(\prod_{i \in I} M_i) / \sim_U$
• Embeds each factor model elementarily into ultraproduct
• Cardinality bounds: $|M| \leq |\prod_{i \in I} M_i / U| \leq |M|^{|I|}$
• Preserves algebraic operations and relations
• Often yields more saturated result than factor models

Łoś's theorem for ultraproducts

• For first-order formula $\phi(x_1, ..., x_n)$ and ultraproduct elements $a_1, ..., a_n$: $\prod_{i \in I} M_i / U \models \phi(a_1, ..., a_n)$ if and only if ${i \in I : M_i \models \phi(a_{1i}, ..., a_{ni})} \in U$
• Transfers first-order properties from factors to ultraproduct
• Proves elementarity of natural embedding of factors into ultraproduct
• Constructs models with specific properties
• Analyzes formula behavior across different models

Ultraproducts in logic theorems

• Compactness theorem: Set of first-order sentences has model if every finite subset has model
  1. Construct models for each finite subset
  2. Form ultraproduct of these models
  3. Use Łoś's theorem to show ultraproduct models all sentences
• Completeness theorem: Sentence provable if and only if true in all models
  • Ultraproducts construct canonical models
  • Show consistent sentence sets have models
• Compactness follows from completeness in first-order logic
• Ultraproducts offer model-theoretic approach to both theorems

Applications of ultraproducts

• Non-standard analysis constructs hyperreal numbers using real number ultraproducts formalizing infinitesimals and infinite numbers
• Ax-Kochen theorem compares p-adic fields of different characteristics applying to number theory and algebraic geometry
• Keisler-Shelah isomorphism theorem: Models elementarily equivalent if and only if isomorphic ultrapowers exist
• Preservation theorems characterize formulas preserved under model-theoretic operations (Łoś-Tarski preservation theorem for substructures)
• Counterexample construction builds models with specific properties (non-standard arithmetic models)