Limit ordinal

Definition

A limit ordinal is an ordinal number that is not zero and cannot be reached by adding 1 (or any finite number) to a smaller ordinal. Instead, it is defined as the least upper bound of all smaller ordinals. Limit ordinals are essential in understanding recursive ordinals and help establish a hierarchy within the broader framework of ordinals and well-orderings.

5 Must Know Facts For Your Next Test

1. Limit ordinals are always greater than all smaller ordinals but cannot be obtained by adding any finite amount to an existing ordinal.
2. The smallest limit ordinal is $\\omega$, which is the first infinite ordinal, representing the set of all finite ordinals.
3. Limit ordinals play a key role in constructing larger ordinals and are fundamental in defining transfinite sequences.
4. Every limit ordinal can be expressed as the supremum (least upper bound) of the set of all smaller ordinals.
5. In the context of well-orderings, limit ordinals help define and distinguish between different levels of infinity and their properties.

Review Questions

• How do limit ordinals differ from finite and successor ordinals, and why are they important in understanding the structure of ordinals?
  • Limit ordinals differ from finite ordinals because they cannot be reached through simple addition; instead, they are defined as the least upper bounds of smaller ordinals. Successor ordinals are those that can be reached by adding one to another ordinal. Limit ordinals are important because they provide crucial insights into the hierarchy and properties of ordinal numbers, showing how infinite sequences can be constructed and understood within this framework.
• Discuss how limit ordinals relate to recursive ordinals and their significance in recursion theory.
  • Limit ordinals are closely related to recursive ordinals as they serve as critical stepping stones in the construction and understanding of recursive functions. Recursive ordinals can sometimes only reach limit ordinals through complex processes. This relationship illustrates how limit ordinals contribute to defining what can be computed or defined recursively, highlighting their importance in both set theory and recursion theory.
• Evaluate the implications of limit ordinals in establishing well-orderings and how they affect our understanding of different sizes of infinity.
  • Limit ordinals have significant implications for establishing well-orderings since they ensure that every non-empty subset of an ordinal has a least element. This property allows for better categorization of infinite sets, contributing to our understanding of different sizes of infinity. By utilizing limit ordinals in this context, we gain insight into how larger sets can be systematically ordered, laying the groundwork for deeper explorations into cardinality and transfinite arithmetic.

Related terms

Ordinal Number: An ordinal number is a way to represent the position or order of elements in a well-ordered set, extending beyond finite numbers to include infinite sequences.

Recursive Ordinal: A recursive ordinal is an ordinal that can be reached by a recursive function, often used to understand the structure of ordinals within recursion theory.

Well-Ordering: Well-ordering is a property of a set that ensures every non-empty subset has a least element, which is crucial for the definition and understanding of ordinals.