Admissible Ordinals

Definition

Admissible ordinals are ordinals that reflect a certain level of complexity in recursive functions and theories, acting as benchmarks for the strength of various logical systems. These ordinals can be used to characterize certain models of set theory and provide insights into the nature of definability and computability. Their significance extends to defining hyperarithmetical sets and understanding recursive pseudo-well-orderings.

5 Must Know Facts For Your Next Test

1. Admissible ordinals are closely tied to the hierarchy of recursive functions, where each ordinal corresponds to a specific level of complexity.
2. The smallest admissible ordinal is known as $eta_0$, which serves as a foundational element in the study of recursive theories.
3. These ordinals can be utilized to construct models of set theory, allowing for the analysis of properties like consistency and completeness.
4. In the context of the hyperarithmetical hierarchy, admissible ordinals help delineate the boundaries between different classes of definable sets.
5. Recursive pseudo-well-orderings are often characterized using admissible ordinals, establishing connections between recursion theory and set-theoretic concepts.

Review Questions

• How do admissible ordinals relate to the complexity levels in the hierarchy of recursive functions?
  • Admissible ordinals serve as indicators of complexity within the hierarchy of recursive functions, where each ordinal represents a specific level. As you move up this hierarchy, the ordinals reflect increasingly complex recursive definitions. This relationship helps in classifying sets and functions based on their computational properties, making admissible ordinals essential for understanding recursion theory.
• Discuss the role of admissible ordinals in defining hyperarithmetical sets and their importance in set theory.
  • Admissible ordinals play a crucial role in defining hyperarithmetical sets, as they help establish boundaries between different classes within this hierarchy. By using these ordinals, one can classify sets based on their definability and computability characteristics. This classification enhances our understanding of set theory and shows how different levels of complexity can be managed through logical frameworks.
• Evaluate the impact of admissible ordinals on recursive pseudo-well-orderings and their implications for computability theory.
  • Admissible ordinals significantly impact recursive pseudo-well-orderings by providing a structured way to understand the ordering of complex sets under recursion. They facilitate the identification of patterns and relationships among recursively defined entities. The implications extend to computability theory, where these orderings help determine how various computational processes interact and relate to one another, ultimately enriching our understanding of recursion and definability.

Related terms

Recursive Functions: Functions that can be computed by a Turing machine or defined using a process of recursion, playing a key role in computability theory.

Ordinal Notation: A system for representing ordinals using sequences or other structured forms, facilitating their manipulation and comparison.

Definable Sets: Sets that can be described by a logical formula within a given language, impacting how we understand computability and admissibility.