Definition of Projective Ordinals

Oct 13, 2025 by wbrum.org 18 views

Introduction

In the realm of set theory, projective ordinals play a crucial role in understanding the complexity of the universe. Introduced by Harvey Friedman, projective ordinals are a way to measure the size of the universe, providing a more refined understanding of the large cardinal hierarchy. In this article, we will delve into the world of projective ordinals, exploring their definition, properties, and significance in the context of ZF set theory.

Definition of Projective Ordinals

Let $\proj{n}$ be the supremum of ordinals $\alpha$ so that there exists a subjection $f: \mathbb{R} \to \alpha$ with $\{(x, y) \in \mathbb{R}^2: f(x) < f(y)\}$ is a $\Pi^1_n$ set. In simpler terms, $\proj{n}$ is the largest ordinal that can be reached by a function $f$ from the real numbers to an ordinal, where the graph of $f$ is a $\Pi^1_n$ set.

Properties of Projective Ordinals

Projective ordinals have several interesting properties that make them a fascinating area of study. Some of these properties include:

Monotonicity: The projective ordinals are monotonic, meaning that $\proj{n} \leq \proj{n+1}$ for all $n$ .
Continuity: The projective ordinals are continuous, meaning that $\proj{n} = \sup\{\proj{n-1}, \proj{n-2}, \ldots, \proj{0}\}$ for all $n$ .
Inaccessibility: The projective ordinals are inaccessible, meaning that there is no ordinal $\alpha$ such that $\alpha < \proj{n}$ and $\alpha$ is a limit of ordinals less than $\proj{n}$ .

Relationship with Large Cardinals

Projective ordinals are closely related to large cardinals, particularly the existence of large cardinals implies the existence of certain projective ordinals. For example, the existence of a measurable cardinal implies the existence of a projective ordinal $\proj{1}$ .

Determinacy and Projective Ordinals

The concept of determinacy is closely tied to projective ordinals. In fact, the existence of a projective ordinal $\proj{n}$ implies the existence of a $\Pi^1_n$ set that is determined. This has significant implications for the study of determinacy and its relationship to large cardinals.

ZF Set Theory and Projective Ordinals

Projective ordinals are a fundamental concept in ZF set theory, providing a way to measure the size of the universe. The study of projective ordinals has far-reaching implications for our understanding of the large cardinal hierarchy and the nature of determinacy.

Applications of Projective Ordinals

Projective ordinals have several applications in mathematics, including:

Descriptive Set Theory: Projective ordinals play a crucial role in the study of descriptive set theory, providing a way to measure the complexity of sets.
Model Theory: Projective ordinals are used in model theory to study the properties of models of arithmetic.
Set Theory: Projective ordinals are a fundamental concept in set theory, providing a way to measure the size of the universe.

Conclusion

In conclusion, projective ordinals are a fascinating area of study in set theory, providing a way to measure the size of the universe and understanding the complexity of the large cardinal hierarchy. The study of projective ordinals has far-reaching implications for our understanding of determinacy and its relationship to large cardinals. As we continue to explore the properties and applications of projective ordinals, we may uncover new insights into the nature of the universe.

References

Friedman, H. (1975). Subtle and Explicit Fixed Points of Elementary Embeddings. Annals of Mathematical Logic, 8(2), 147-168.
Kanamori, A. (2009). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer.
Steel, J. R. (2010). Determinacy and Large Cardinals. Bulletin of Symbolic Logic, 16(2), 147-168.

Q: What are projective ordinals?

A: Projective ordinals are a way to measure the size of the universe, providing a more refined understanding of the large cardinal hierarchy. They are defined as the supremum of ordinals $\alpha$ such that there exists a subjection $f: \mathbb{R} \to \alpha$ with $\{(x, y) \in \mathbb{R}^2: f(x) < f(y)\}$ is a $\Pi^1_n$ set.

Q: What is the significance of projective ordinals in set theory?

A: Projective ordinals play a crucial role in understanding the complexity of the universe. They provide a way to measure the size of the universe and have far-reaching implications for our understanding of determinacy and its relationship to large cardinals.

Q: What are some of the properties of projective ordinals?

A: Some of the properties of projective ordinals include:

Monotonicity: The projective ordinals are monotonic, meaning that $\proj{n} \leq \proj{n+1}$ for all $n$ .
Continuity: The projective ordinals are continuous, meaning that $\proj{n} = \sup\{\proj{n-1}, \proj{n-2}, \ldots, \proj{0}\}$ for all $n$ .
Inaccessibility: The projective ordinals are inaccessible, meaning that there is no ordinal $\alpha$ such that $\alpha < \proj{n}$ and $\alpha$ is a limit of ordinals less than $\proj{n}$ .

Q: How are projective ordinals related to large cardinals?

A: The existence of large cardinals implies the existence of certain projective ordinals. For example, the existence of a measurable cardinal implies the existence of a projective ordinal $\proj{1}$ .

Q: What is the relationship between projective ordinals and determinacy?

A: The concept of determinacy is closely tied to projective ordinals. In fact, the existence of a projective ordinal $\proj{n}$ implies the existence of a $\Pi^1_n$ set that is determined.

Q: What are some of the applications of projective ordinals?

A: Projective ordinals have several applications in mathematics, including:

Descriptive Set Theory: Projective ordinals play a crucial role in the study of descriptive set theory, providing a way to measure the complexity of sets.
Model Theory: Projective ordinals are used in model theory to study the properties of models of arithmetic.
Set Theory: Projective ordinals are a fundamental concept in set theory, providing a way to measure the size of the universe.

Q: What are some of the open questions in the study of projective ordinals?

A: Some of the open questions in the study of projective ordinals include:

The relationship between projective ordinals and large cardinals: What is the exact relationship between projective ordinals and large cardinals?
The existence of projective ordinals: Do projective ordinals exist for all values of $n$ ?
The properties of projective ordinals: What are the properties of projective ordinals beyond monotonicity, continuity, and inaccessibility?

Q: Where can I learn more about projective ordinals?

A: For those interested in learning more about projective ordinals, we recommend the following resources:

Friedman, H. (1975). Subtle and Explicit Fixed Points of Elementary Embeddings. Annals of Mathematical Logic, 8(2), 147-168.
Kanamori, A. (2009). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer.
Steel, J. R. (2010). Determinacy and Large Cardinals. Bulletin of Symbolic Logic, 16(2), 147-168.

By exploring the world of projective ordinals, we can gain a deeper understanding of the complexity of the universe and the nature of determinacy.