Successor Ordinal

Definition

A successor ordinal is an ordinal number that comes immediately after another ordinal in the ordering of ordinals. It is formed by taking an existing ordinal and adding one to it, representing the next step in the well-ordered set of ordinals. Successor ordinals play a crucial role in understanding the hierarchy of ordinals and how they are structured, particularly when considering their properties in relation to limit ordinals and well-orderings.

5 Must Know Facts For Your Next Test

1. Every finite ordinal is a successor ordinal except for 0, which is the smallest ordinal.
2. Successor ordinals can be denoted as $$eta + 1$$ where $$eta$$ is an existing ordinal.
3. In the context of infinite ordinals, for any ordinal $$eta$$, the successor ordinal $$eta + 1$$ is always greater than $$eta$$.
4. The set of all successor ordinals is always dense within the set of all ordinals, meaning there are infinitely many ordinals between any two ordinals.
5. Successor ordinals are essential for constructing ordinal arithmetic and understanding the properties of ordinals in set theory.

Review Questions

• How do successor ordinals relate to limit ordinals in the hierarchy of ordinals?
  • Successor ordinals represent the immediate next step in the ordering of ordinals, while limit ordinals represent a collection of smaller ordinals that do not follow an immediate predecessor. For instance, while 2 is a successor ordinal following 1, the ordinal 3 is a limit ordinal as it encompasses all smaller finite ordinals without having a direct predecessor. Understanding this distinction helps clarify how different types of ordinals interact within well-ordered sets.
• Discuss the implications of successor ordinals on the well-ordering principle.
  • Successor ordinals reinforce the well-ordering principle by ensuring that every non-empty set of ordinals has a least element. Since every finite ordinal can be expressed as a successor ordinal (except for 0), this property confirms that even when dealing with infinite collections of ordinals, there will always be a defined 'next' step in terms of ordering. This supports structured approaches in set theory and shows how orderings can be consistently maintained.
• Evaluate how understanding successor ordinals contributes to grasping concepts in set theory and recursive functions.
  • Understanding successor ordinals is critical for grasping more complex concepts within set theory and recursive functions. They form the basis for defining operations on ordinals and provide insight into ordinal arithmetic, which is foundational for building further mathematical structures. By evaluating how successor ordinals function in relation to limit ordinals and well-orderings, one can appreciate their role in recursive definitions and algorithms, leading to deeper insights into mathematical reasoning and problem-solving strategies.

Related terms

Limit Ordinal: A limit ordinal is an ordinal that is not a successor ordinal; it cannot be reached by adding one to any smaller ordinal, representing a 'limit' of smaller ordinals.

Well-Ordering Principle: The well-ordering principle states that every non-empty set of ordinals has a least element, establishing a fundamental property of the ordering of ordinals.

Ordinal Number: An ordinal number is a generalization of natural numbers used to describe the order type of well-ordered sets, representing positions or ranks.