8.1 Ultrafilters and their properties

Ultrafilters are powerful tools in model theory, maximizing filter properties and containing exactly one of A or its complement for any subset A. They're crucial for building ultraproducts, which help analyze complex mathematical structures and preserve certain properties across related objects.

Non-principal ultrafilters, containing no finite sets, exist on infinite sets but require the Axiom of Choice. They're key in proving Łoś's theorem, studying saturation, and constructing nonstandard models. Ultrafilters bridge set theory, topology, and analysis, revealing deep connections in mathematics.

Ultrafilters in Model Theory

Definition and Key Properties

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• Ultrafilters maximize filter properties on a set containing exactly one of A or its complement for any subset A
• Collection of subsets U on set X satisfies three properties
  • U non-empty
  • U closed under supersets
  • Intersection of any two sets in U also in U
• Play crucial role allowing construction of ultraproducts fundamental tools for building and analyzing mathematical structures
• Used to "average" family of structures preserving certain properties and creating new models with desired characteristics

Applications in Model Theory

• Essential in proving Łoś's theorem relating truth of first-order sentences in ultraproduct to factor structures
• Extend to study of saturation elementary equivalence and construction of nonstandard models
• Enable analysis of complex mathematical structures through ultraproduct construction
• Facilitate preservation and transfer of properties between related mathematical objects

Existence of Non-Principal Ultrafilters

Zorn's Lemma and Non-Principal Ultrafilters

• Non-principal ultrafilters contain no finite sets existence not constructively provable in ZF set theory without additional axioms
• Zorn's Lemma states every partially ordered set with upper bound for every chain contains maximal element (equivalent to Axiom of Choice)
• Proof starts with cofinite filter on infinite set X (subsets of X with finite complements)
• Define partial order on set of filters extending cofinite filter ordered by inclusion
• Apply Zorn's Lemma show partially ordered set has maximal element necessarily an ultrafilter
• Resulting ultrafilter non-principal extends cofinite filter cannot contain finite set

Significance and Implications

• Demonstrates power of Zorn's Lemma establishing existence of mathematical objects without explicit construction
• Highlights connection between set-theoretic axioms and existence of certain mathematical structures
• Reveals limitations of constructive mathematics in certain areas of set theory and model theory
• Provides foundation for studying properties of non-principal ultrafilters in various mathematical contexts (topology analysis)

Properties of Ultrafilters

Closure and Intersection Properties

• Possess all filter properties with additional maximality conditions
• Closure under supersets If A in ultrafilter U and A ⊆ B then B in U (ensures any set containing set in ultrafilter also in ultrafilter)
• Finite intersection property If A and B in U then A ∩ B in U (extends to any finite number of sets)
• Maximality property For any subset A of base set X either A or complement X\A in ultrafilter not both
• Prime filters If A ∪ B in U then A in U or B in U (or both)
• Intersection of all sets in non-principal ultrafilter empty
• Finite intersection property (FIP) intersection of any finite subcollection of sets in ultrafilter non-empty

Advanced Properties and Characterizations

• Ultrafilters characterized as maximal filters cannot be properly extended to larger filter
• Every ultrafilter on finite set principal generated by single element
• On infinite set non-principal ultrafilters exist but require Axiom of Choice for proof
• Ultrafilters preserve Boolean operations intersection union and complement in specific ways
• Ultrafilters on Boolean algebras correspond to homomorphisms from algebra to two-element Boolean algebra

Principal vs Non-Principal Ultrafilters

Definitions and Examples

• Principal ultrafilters contain smallest element typically singleton set {x} consist of all supersets
• Non-principal ultrafilters contain no finite sets not generated by single element of base set
• Example principal ultrafilter On natural numbers N collection of all subsets containing fixed number n
• Example non-principal ultrafilter On N contains all cofinite sets (complements finite) and additional sets chosen to satisfy ultrafilter properties
• Principal ultrafilters exist on any set non-principal require infinite set and typically rely on Axiom of Choice

Significance in Mathematical Applications

• Non-principal ultrafilters used in construction of nonstandard models of arithmetic
• Applied in study of convergence in topological spaces
• Distinction crucial in construction of ultraproducts and study of compactifications in topology
• Principal ultrafilters correspond to points in Stone–Čech compactification non-principal to "points at infinity"
• Non-principal ultrafilters enable construction of hyperreal numbers in non-standard analysis
• Used in functional analysis to study convergence properties of sequences and nets