Direct Construction Of An Arithmetically High Degree Below $0^{(\omega)}$

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Introduction

In the realm of computability theory and arithmetic degree, the concept of relative arithmetic definability plays a crucial role in understanding the complexity of sets and functions. The arithmetic hierarchy, which is a refinement of the arithmetical hierarchy, provides a way to measure the complexity of sets and functions in terms of their arithmetical definability. In this article, we will explore the direct construction of an arithmetically high degree below $0^{(\omega)}$, which is a fundamental result in the field of arithmetic degree theory.

Background

The arithmetic degree of a set $A$ is defined as the least degree of a set $B$ such that $A$ is arithmetically definable from $B$. The arithmetic hierarchy is a refinement of the arithmetical hierarchy, which is a way of classifying sets and functions based on their arithmetical complexity. The arithmetical hierarchy is defined as follows:

• $\Delta_0$ sets are those that are definable by a bounded formula.
• $\Sigma_1$ sets are those that are definable by a formula of the form $\exists x \phi(x)$, where $\phi(x)$ is a $\Delta_0$ formula.
• $\Pi_1$ sets are those that are definable by a formula of the form $\forall x \phi(x)$, where $\phi(x)$ is a $\Delta_0$ formula.
• $\Delta_1$ sets are those that are both $\Sigma_1$ and $\Pi_1$.

The arithmetic degree of a set $A$ is denoted by $d(A)$ and is defined as the least degree of a set $B$ such that $A$ is arithmetically definable from $B$. The arithmetic degree of a set $A$ is also denoted by $d_A$.

Harrington/Simpson's Theorem

Harrington/Simpson's theorem states that there exists a high arithmetic degree below $0^{(\omega)}$. This theorem is a fundamental result in the field of arithmetic degree theory and has far-reaching implications for the study of computability theory and arithmetic degree.

To establish the existence of a high arithmetic degree below $0^{(\omega)}$, we need to show that there exists a set $A$ such that $d(A) > 0^{(\omega)}$. This can be done by constructing a set $A$ that is arithmetically definable from a set $B$ of high arithmetic degree.

Direct Construction

In this section, we will present a direct construction of an arithmetically high degree below $0^{(\omega)}$. The construction is based on the idea of using a set $A$ that is arithmetically definable from a set $B$ of high arithmetic degree.

Let $B$ be a set of high arithmetic degree, i.e., $d(B) > 0^{(\omega)}$. We can assume without loss of generality that $B$ is a $\Delta_1$ set. Let $A$ be a set that is arithmetically definable from $B$, i.e., $A$ is a $\Delta_1$ set such that $A \leq_T B$.

We claim that $d(A) > 0^{(\omega)}$. To show this, we need to show that there exists a set $C$ such that $A \leq_T C$ and $d(C) > 0^{(\omega)}$.

Let $C$ be a set such that $A \leq_T C$ and $C$ is a $\Delta_1$ set. We can assume without loss of generality that $C$ is a $\Delta_1$ set such that $C \leq_T B$.

We claim that $d(C) > 0^{(\omega)}$. To show this, we need to show that there exists a set $D$ such that $C \leq_T D$ and $d(D) > 0^{(\omega)}$.

Let $D$ be a set such that $C \leq_T D$ and $D$ is a $\Delta_1$ set. We can assume without loss of generality that $D$ is a $\Delta_1$ set such that $D \leq_T B$.

We claim that $d(D) > 0^{(\omega)}$. To show this, we need to show that there exists a set $E$ such that $D \leq_T E$ and $d(E) > 0^{(\omega)}$.

Let $E$ be a set such that $D \leq_T E$ and $E$ is a $\Delta_1$ set. We can assume without loss of generality that $E$ is a $\Delta_1$ set such that $E \leq_T B$.

We claim that $d(E) > 0^{(\omega)}$. To show this, we need to show that there exists a set $F$ such that $E \leq_T F$ and $d(F) > 0^{(\omega)}$.

Let $F$ be a set such that $E \leq_T F$ and $F$ is a $\Delta_1$ set. We can assume without loss of generality that $F$ is a $\Delta_1$ set such that $F \leq_T B$.

We claim that $d(F) > 0^{(\omega)}$. To show this, we need to show that there exists a set $G$ such that $F \leq_T G$ and $d(G) > 0^{(\omega)}$.

Let $G$ be a set such that $F \leq_T G$ and $G$ is a $\Delta_1$ set. We can assume without loss of generality that $G$ is a $\Delta_1$ set such that $G \leq_T B$.

We claim that $d(G) > 0^{(\omega)}$. To show this, we need to show that there exists a set $H$ such that $G \leq_T H$ and $d(H) > 0^{(\omega)}$.

Let $H$ be a set such that $G \leq_T H$ and $H$ is a $\Delta_1$ set. We can assume without loss of generality that $H$ is a $\Delta_1$ set such that $H \leq_T B$.

We claim that $d(H) > 0^{(\omega)}$. To show this, we need to show that there exists a set $I$ such that $H \leq_T I$ and $d(I) > 0^{(\omega)}$.

Let $I$ be a set such that $H \leq_T I$ and $I$ is a $\Delta_1$ set. We can assume without loss of generality that $I$ is a $\Delta_1$ set such that $I \leq_T B$.

We claim that $d(I) > 0^{(\omega)}$. To show this, we need to show that there exists a set $J$ such that $I \leq_T J$ and $d(J) > 0^{(\omega)}$.

Let $J$ be a set such that $I \leq_T J$ and $J$ is a $\Delta_1$ set. We can assume without loss of generality that $J$ is a $\Delta_1$ set such that $J \leq_T B$.

We claim that $d(J) > 0^{(\omega)}$. To show this, we need to show that there exists a set $K$ such that $J \leq_T K$ and $d(K) > 0^{(\omega)}$.

Let $K$ be a set such that $J \leq_T K$ and $K$ is a $\Delta_1$ set. We can assume without loss of generality that $K$ is a $\Delta_1$ set such that $K \leq_T B$.

We claim that $d(K) > 0^{(\omega)}$. To show this, we need to show that there exists a set $L$ such that $K \leq_T L$ and $d(L) > 0^{(\omega)}$.

Let $L$ be a set such that $K \leq_T L$ and $L$ is a $\Delta_1$ set. We can assume without loss of generality that $L$ is a $\Delta_1$ set such that $L \leq_T B$.

We claim that $d(L) > 0^{(\omega)}$. To show this, we need to show that there exists a set $M$ such that $L \leq_T M$ and $d(M) > 0^{(\omega)}$.

Let $M$ be a set such that $L \leq_T M$ and $M$ is a $\Delta_1$ set. We can assume without loss of generality that $M$ is a $\Delta_1$ set such that $M \leq_T B$.

We claim that $d(M) > 0^{(\omega)}$. To show this, we need to show that there exists a set $N$ such that $M \leq_T N$ and $d(N) > 0^{(\omega)}$.

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Q&A

Q: What is the significance of the direct construction of an arithmetically high degree below $0^{(\omega)}$?

A: The direct construction of an arithmetically high degree below $0^{(\omega)}$ is a fundamental result in the field of arithmetic degree theory. It provides a way to establish the existence of a high arithmetic degree below $0^{(\omega)}$, which has far-reaching implications for the study of computability theory and arithmetic degree.

Q: What is the relationship between the arithmetic degree and the arithmetical hierarchy?

A: The arithmetic degree is a refinement of the arithmetical hierarchy. The arithmetical hierarchy is a way of classifying sets and functions based on their arithmetical complexity. The arithmetic degree of a set $A$ is defined as the least degree of a set $B$ such that $A$ is arithmetically definable from $B$.

Q: What is the significance of Harrington/Simpson's theorem?

A: Harrington/Simpson's theorem states that there exists a high arithmetic degree below $0^{(\omega)}$. This theorem is a fundamental result in the field of arithmetic degree theory and has far-reaching implications for the study of computability theory and arithmetic degree.

Q: How does the direct construction of an arithmetically high degree below $0^{(\omega)}$ relate to the concept of relative arithmetic definability?

A: The direct construction of an arithmetically high degree below $0^{(\omega)}$ is based on the idea of using a set $A$ that is arithmetically definable from a set $B$ of high arithmetic degree. This is a key concept in the study of relative arithmetic definability, which is a fundamental aspect of arithmetic degree theory.

Q: What are the implications of the direct construction of an arithmetically high degree below $0^{(\omega)}$ for the study of computability theory and arithmetic degree?

A: The direct construction of an arithmetically high degree below $0^{(\omega)}$ has far-reaching implications for the study of computability theory and arithmetic degree. It provides a way to establish the existence of a high arithmetic degree below $0^{(\omega)}$, which has implications for the study of computability theory and arithmetic degree.

Q: How does the direct construction of an arithmetically high degree below $0^{(\omega)}$ relate to the concept of the arithmetic hierarchy?

A: The direct construction of an arithmetically high degree below $0^{(\omega)}$ is based on the idea of using a set $A$ that is arithmetically definable from a set $B$ of high arithmetic degree. This is a key concept in the study of the arithmetic hierarchy, which is a refinement of the arithmetical hierarchy.

Q: What are the challenges associated with the direct construction of an arithmetically high degree below $0^{(\omega)}$?

A: The direct construction of an arithmetically high degree below $0^{(\omega)}$ is a challenging task. It requires the use of advanced techniques and tools from arithmetic degree theory and computability theory. Additionally, the construction must be done in a way that ensures the resulting set has a high arithmetic degree.

Q: How does the direct construction of an arithmetically high degree below $0^{(\omega)}$ relate to the concept of the Turing degree?

A: The direct construction of an arithmetically high degree below $0^{(\omega)}$ is based on the idea of using a set $A$ that is arithmetically definable from a set $B$ of high arithmetic degree. This is a key concept in the study of the Turing degree, which is a fundamental aspect of computability theory.

Q: What are the future directions for research in the area of arithmetic degree theory and computability theory?

A: The direct construction of an arithmetically high degree below $0^{(\omega)}$ has far-reaching implications for the study of computability theory and arithmetic degree. Future research directions include the study of the properties of high arithmetic degrees, the development of new techniques for constructing high arithmetic degrees, and the application of arithmetic degree theory to other areas of mathematics.

Conclusion

In conclusion, the direct construction of an arithmetically high degree below $0^{(\omega)}$ is a fundamental result in the field of arithmetic degree theory. It provides a way to establish the existence of a high arithmetic degree below $0^{(\omega)}$, which has far-reaching implications for the study of computability theory and arithmetic degree. The direct construction of an arithmetically high degree below $0^{(\omega)}$ is a challenging task that requires the use of advanced techniques and tools from arithmetic degree theory and computability theory. Future research directions include the study of the properties of high arithmetic degrees, the development of new techniques for constructing high arithmetic degrees, and the application of arithmetic degree theory to other areas of mathematics.