9.3 Relationship between hyperarithmetical and Δ^1_1 sets

Hyperarithmetical sets and Δ^1_1 sets are crucial in recursive function theory and computability. They provide a hierarchical classification of sets based on complexity and definability, bridging the arithmetic and analytical hierarchies.

Understanding the relationship between these sets is key to studying the analytical hierarchy's structure. While every hyperarithmetical set is Δ^1_1, the reverse isn't always true. This distinction has significant implications for computability and decidability in mathematics.

Hyperarithmetical sets

• Hyperarithmetical sets play a crucial role in the study of recursive functions and computability theory
• Provide a hierarchical classification of sets based on their complexity and definability using recursive functions
• Serve as a bridge between the arithmetic hierarchy and the analytical hierarchy

Definition of hyperarithmetical sets

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• A set $A \subseteq \mathbb{N}$ is hyperarithmetical if it is $\Delta^1_1$ definable in the language of second-order arithmetic
• Equivalently, a set is hyperarithmetical if it is computable from Kleene's $\mathcal{O}$, a $\Pi^1_1$-complete set
• Hyperarithmetical sets can be defined using recursive transfinite progressions of the arithmetic hierarchy up to the ordinal $\omega_1^{CK}$, the first non-recursive ordinal

Closure properties of hyperarithmetical sets

• The collection of hyperarithmetical sets is closed under complement, union, and intersection
• If $A$ and $B$ are hyperarithmetical sets, then so are $A \cup B$, $A \cap B$, and $\overline{A}$
• Hyperarithmetical sets are closed under recursive substitution and bounded quantification
• The collection of hyperarithmetical sets forms a $\sigma$-algebra, a collection of sets closed under countable unions and complements

Δ^1_1 sets

• $\Delta^1_1$ sets are an important class of sets in descriptive set theory and recursive function theory
• They provide a natural extension of the arithmetic hierarchy to the analytical hierarchy
• $\Delta^1_1$ sets are closely related to the notion of hyperarithmeticity and have significant implications for computability and definability

Definition of Δ^1_1 sets

• A set $A \subseteq \mathbb{N}$ is $\Delta^1_1$ if it is both $\Sigma^1_1$ and $\Pi^1_1$
• $\Sigma^1_1$ sets are definable by a formula of the form $\exists X \varphi(n, X)$, where $\varphi$ is an arithmetic formula and $X$ is a set variable
• $\Pi^1_1$ sets are definable by a formula of the form $\forall X \varphi(n, X)$, where $\varphi$ is an arithmetic formula and $X$ is a set variable

Closure properties of Δ^1_1 sets

• The collection of $\Delta^1_1$ sets is closed under complement, union, and intersection
• If $A$ and $B$ are $\Delta^1_1$ sets, then so are $A \cup B$, $A \cap B$, and $\overline{A}$
• $\Delta^1_1$ sets are closed under recursive substitution and bounded quantification
• The collection of $\Delta^1_1$ sets forms a $\sigma$-algebra, a collection of sets closed under countable unions and complements

Relationship between hyperarithmetical and Δ^1_1 sets

• Hyperarithmetical sets and $\Delta^1_1$ sets are closely related and share many properties
• Understanding the relationship between these two classes of sets is crucial for studying the structure of the analytical hierarchy and its implications for computability

Hyperarithmetical sets as subsets of Δ^1_1 sets

• Every hyperarithmetical set is also a $\Delta^1_1$ set
• The collection of hyperarithmetical sets is a proper subset of the collection of $\Delta^1_1$ sets
• There exist $\Delta^1_1$ sets that are not hyperarithmetical, such as the set of indices of recursive well-orderings of $\mathbb{N}$

Δ^1_1 sets as extensions of hyperarithmetical sets

• $\Delta^1_1$ sets can be seen as a natural extension of hyperarithmetical sets to the analytical hierarchy
• While hyperarithmetical sets are defined using recursive transfinite progressions up to $\omega_1^{CK}$, $\Delta^1_1$ sets allow for definitions using set quantification
• The jump operator, which is used to define the arithmetic hierarchy, can be extended to the hyperjump operator to define the hyperarithmetical hierarchy

Differences in closure properties

• Both hyperarithmetical sets and $\Delta^1_1$ sets are closed under complement, union, intersection, recursive substitution, and bounded quantification
• However, $\Delta^1_1$ sets have additional closure properties, such as closure under recursive $\omega$-unions and $\omega$-intersections
• The collection of $\Delta^1_1$ sets is also closed under the $\Delta^1_1$-jump operator, while the collection of hyperarithmetical sets is not

Implications for computability and decidability

• The relationship between hyperarithmetical sets and $\Delta^1_1$ sets has significant implications for the study of computability and decidability
• Hyperarithmetical sets are $\Delta^1_1$-computable, meaning they can be computed by a $\Delta^1_1$-recursive function
• The set of indices of $\Delta^1_1$ sets is $\Pi^1_1$-complete, implying that many questions about $\Delta^1_1$ sets are undecidable

Applications of hyperarithmetical and Δ^1_1 sets

• Hyperarithmetical sets and $\Delta^1_1$ sets have numerous applications in various areas of mathematical logic and computability theory
• These applications highlight the importance of understanding the properties and relationships of these classes of sets

Role in recursive function theory

• Hyperarithmetical sets and $\Delta^1_1$ sets are fundamental in the study of recursive functions and computability
• They provide a framework for classifying sets based on their complexity and definability using recursive functions
• The hyperarithmetical hierarchy and the $\Delta^1_1$-jump operator are essential tools for analyzing the structure of the recursive functions

Significance in mathematical logic

• The study of hyperarithmetical sets and $\Delta^1_1$ sets is closely related to the foundations of mathematics and mathematical logic
• These classes of sets are important in the study of second-order arithmetic and the analytical hierarchy
• Understanding the properties of these sets contributes to the development of proof theory, model theory, and set theory

Connections to other areas of mathematics

• Hyperarithmetical sets and $\Delta^1_1$ sets have connections to various other areas of mathematics, such as topology, analysis, and algebra
• In topology, the Borel hierarchy and the projective hierarchy are related to the analytical hierarchy and the study of $\Delta^1_1$ sets
• In analysis, the study of recursive functions and $\Delta^1_1$ sets is connected to the theory of computable analysis and constructive mathematics
• In algebra, the study of recursive structures and computable algebraic structures often involves the use of hyperarithmetical sets and $\Delta^1_1$ sets