🧠Model Theory Unit 8 – Ultraproducts and Ultrapowers

Key Concepts and Definitions

• Ultraproducts generalize the notion of direct products in model theory by using an ultrafilter to determine which elements are "large" or "significant"
• An ultrafilter $\mathcal{U}$ on a set $I$ is a collection of subsets of $I$ that is closed under finite intersections, contains $I$, and for every subset $A \subseteq I$, either $A \in \mathcal{U}$ or $I \setminus A \in \mathcal{U}$
  • Ultrafilters can be principal (generated by a single element) or non-principal (free ultrafilters)
• Given a family of structures $\{M_i\}_{i \in I}$ and an ultrafilter $\mathcal{U}$ on $I$, the ultraproduct $\prod_{i \in I} M_i/\mathcal{U}$ is the quotient of the direct product by the equivalence relation induced by $\mathcal{U}$
• Ultrapowers are a special case of ultraproducts where all the structures in the family are the same ($M_i = M$ for all $i \in I$)
• The canonical embedding of a structure $M$ into its ultrapower $M^I/\mathcal{U}$ is given by the diagonal map $d: M \to M^I/\mathcal{U}$, where $d(a) = [(a)_{i \in I}]_\mathcal{U}$
• Elementary equivalence ($\equiv$) is a central notion in model theory, and ultraproducts preserve elementary equivalence between structures

Filters and Ultrafilters

• A filter $\mathcal{F}$ on a set $I$ is a collection of subsets of $I$ that is closed under finite intersections, closed under supersets, and does not contain the empty set
• Filters can be partially ordered by inclusion, and maximal filters with respect to this ordering are called ultrafilters
• The principal filter generated by an element $i \in I$ is the collection $\mathcal{F}_i = \{A \subseteq I : i \in A\}$
• Every filter can be extended to an ultrafilter using Zorn's Lemma or the Axiom of Choice
• Ultrafilters have the property that for every subset $A \subseteq I$, either $A \in \mathcal{U}$ or $I \setminus A \in \mathcal{U}$, but not both
  • This property is key to the construction and properties of ultraproducts
• The existence of non-principal ultrafilters requires the Axiom of Choice or a weaker form of it

Constructing Ultraproducts

• Given a family of structures $\{M_i\}_{i \in I}$ and an ultrafilter $\mathcal{U}$ on $I$, the ultraproduct $\prod_{i \in I} M_i/\mathcal{U}$ is constructed as follows:
  1. Form the direct product $\prod_{i \in I} M_i$, which consists of all functions $f: I \to \bigcup_{i \in I} M_i$ such that $f(i) \in M_i$ for each $i \in I$
  2. Define an equivalence relation $\sim_\mathcal{U}$ on the direct product by $f \sim_\mathcal{U} g$ if and only if $\{i \in I : f(i) = g(i)\} \in \mathcal{U}$
  3. The ultraproduct is the quotient $\prod_{i \in I} M_i/\mathcal{U} = (\prod_{i \in I} M_i)/\sim_\mathcal{U}$, with elements denoted by $[f]_\mathcal{U}$
• Constants, functions, and relations in the ultraproduct are defined pointwise using representatives from the equivalence classes
  • For example, if $c_i$ is a constant in $M_i$ for each $i \in I$, then the corresponding constant in the ultraproduct is $[(c_i)_{i \in I}]_\mathcal{U}$
• Ultrapowers are constructed similarly, with $M_i = M$ for all $i \in I$, and the ultrapower is denoted by $M^I/\mathcal{U}$

Properties of Ultraproducts

• Ultraproducts preserve first-order properties, meaning that if a first-order sentence holds in "almost all" structures $M_i$ (i.e., the set of indices where it holds is in the ultrafilter), then it holds in the ultraproduct
• The ultraproduct of a family of finite structures can be infinite if the ultrafilter is non-principal
  • This allows for the construction of infinite structures with specific properties
• Ultraproducts commute with reducts, meaning that if $\{M_i\}_{i \in I}$ is a family of $\mathcal{L}$-structures and $\mathcal{L}' \subseteq \mathcal{L}$, then $(\prod_{i \in I} M_i/\mathcal{U})|_{\mathcal{L}'} \cong \prod_{i \in I} (M_i|_{\mathcal{L}'})/\mathcal{U}$
• Ultraproducts preserve elementary equivalence, so if $M_i \equiv N_i$ for all $i \in I$, then $\prod_{i \in I} M_i/\mathcal{U} \equiv \prod_{i \in I} N_i/\mathcal{U}$
• The Fundamental Theorem of Ultraproducts states that an ultraproduct of models of a theory $T$ is itself a model of $T$

Łoś's Theorem and Applications

• Łoś's Theorem is a fundamental result in model theory that relates the truth of first-order formulas in an ultraproduct to their truth in the component structures
  • It states that for any first-order formula $\varphi(x_1, \ldots, x_n)$ and elements $[f_1]_\mathcal{U}, \ldots, [f_n]_\mathcal{U}$ in the ultraproduct, $\prod_{i \in I} M_i/\mathcal{U} \models \varphi([f_1]_\mathcal{U}, \ldots, [f_n]_\mathcal{U})$ if and only if $\{i \in I : M_i \models \varphi(f_1(i), \ldots, f_n(i))\} \in \mathcal{U}$
• Łoś's Theorem has numerous applications in model theory, including:
  • Proving the Compactness Theorem for first-order logic
  • Constructing saturated and homogeneous models
  • Studying the model theory of specific structures (fields, groups, etc.)
• Łoś's Theorem can be used to transfer properties between structures and their ultrapowers, such as:
  • If $M$ is a field, then its ultrapower $M^I/\mathcal{U}$ is also a field
  • If $M$ is a real closed field, then its ultrapower $M^I/\mathcal{U}$ is a real closed field containing $M$

Ultrapowers and Their Significance

• Ultrapowers are a special case of ultraproducts where all the structures in the family are the same ($M_i = M$ for all $i \in I$)
• The canonical embedding of a structure $M$ into its ultrapower $M^I/\mathcal{U}$ is an elementary embedding, preserving all first-order properties
  • This embedding allows for the study of a structure within a larger, saturated extension
• Ultrapowers can be used to construct saturated extensions of structures, which are important in model theory
  • A structure $M$ is $\kappa$-saturated if for every subset $A \subseteq M$ with $|A| < \kappa$ and every type $p(x)$ over $A$, if $p(x)$ is consistent with the theory of $M$, then $p(x)$ is realized in $M$
• Ultrapowers play a crucial role in the study of large cardinals in set theory, such as measurable cardinals and strongly compact cardinals
• The ultrapower construction can be iterated transfinitely to obtain larger extensions and study the model-theoretic properties of structures under such iterations

Advanced Topics and Extensions

• Keisler's Order is a pre-order on complete first-order theories that compares the saturation properties of their ultrapowers
  • Two theories $T_1$ and $T_2$ are said to be Keisler equivalent if for every cardinal $\lambda$, $T_1$ has a $\lambda$-saturated ultrapower if and only if $T_2$ has a $\lambda$-saturated ultrapower
• Shelah's Ultrapower Theorem states that for any infinite structure $M$ in a countable language, there exists a countably incomplete ultrafilter $\mathcal{U}$ such that the ultrapower $M^I/\mathcal{U}$ is $\aleph_1$-saturated
• The notion of ultraproducts can be generalized to reduced products, where the quotient is taken with respect to a filter instead of an ultrafilter
  • Reduced products do not necessarily preserve elementary equivalence but can still be useful in certain model-theoretic constructions
• Ultraproducts and ultrapowers can be defined for many-sorted structures and for structures in infinitary languages, extending their applicability beyond first-order logic
• The study of ultraproducts and ultrapowers has connections to other areas of mathematics, such as algebra (ultraproducts of groups, rings, and modules), topology (ultraproducts of topological spaces), and analysis (ultraproducts of Banach spaces and C*-algebras)

Problem-Solving Techniques

• When working with ultraproducts, it is often helpful to choose representatives carefully from the equivalence classes to simplify computations and proofs
• Łoś's Theorem is a powerful tool for transferring properties between structures and their ultraproducts
  • To prove that an ultraproduct has a certain first-order property, it suffices to show that the set of indices where the component structures have that property is in the ultrafilter
• When dealing with ultrapowers, consider the canonical embedding and use it to relate the structure to its ultrapower
  • Properties that are preserved by elementary embeddings (elementary properties) will hold in the ultrapower
• Use saturation arguments to construct models with specific properties
  • If a type over a small set is consistent with the theory of a structure, then it can be realized in a sufficiently saturated ultrapower of that structure
• Consider the interplay between ultraproducts and other model-theoretic constructions, such as elementary substructures, elementary chains, and homogeneous models
• Analyze the role of the ultrafilter in the properties of the ultraproduct
  • Different ultrafilters can yield ultraproducts with different characteristics, so choosing the appropriate ultrafilter can be crucial in problem-solving
• Utilize the connections between ultraproducts and other areas of mathematics, such as algebra and topology, to gain insights and apply techniques from those fields when appropriate