Turing Degrees Of The Sets Of ZFC-provable Halting/non-halting Programs?

Introduction

In the realm of computability theory, the study of Turing degrees has been a crucial area of research. Turing degrees provide a way to measure the relative computability of sets of natural numbers. In this article, we will explore the Turing degrees of the sets of ZFC-provable halting and non-halting programs. We will define the sets $W_{ZH}$, $W_{ZN}$, and $W_{Z}$, and investigate their Turing degrees.

Background

To begin, let us recall some basic definitions from computability theory. A Turing machine is a mathematical model for computation that consists of a tape divided into cells, each of which can hold a symbol from a finite alphabet. The machine reads and writes symbols on the tape, and its behavior is determined by a set of rules, or a program. A universal Turing machine, denoted by $U$, is a Turing machine that can simulate the behavior of any other Turing machine. In other words, $U$ can take as input the description of any Turing machine and its input, and produce the same output as the original machine.

ZFC-Provable Halting and Non-Halting Programs

Let $U$ be a universal Turing machine. We define the following sets:

• $W_{ZH}$: the set of inputs to $U$ that ZFC-provably halt.
• $W_{ZN}$: the set of inputs to $U$ that ZFC-provably never halt.
• $W_{Z}$: the set of inputs to $U$ that ZFC-provably either halt or never halt.

Here, ZFC refers to the Zermelo-Fraenkel axioms with the axiom of choice, which is a standard foundation for mathematics. A set is said to be ZFC-provable if it can be proved to exist or not exist using the ZFC axioms.

Turing Degrees

A Turing degree is a measure of the relative computability of a set of natural numbers. Two sets are said to have the same Turing degree if they are computably equivalent, meaning that there exists a computable function that can reduce one set to the other. The Turing degrees form a partial order, with the least Turing degree being the degree of the empty set, and the greatest Turing degree being the degree of the set of all natural numbers.

The Turing Degrees of $W_{ZH}$ and $W_{ZN}$

We now investigate the Turing degrees of $W_{ZH}$ and $W_{ZN}$. We claim that $W_{ZH}$ and $W_{ZN}$ have the same Turing degree.

Theorem 1

$W_{ZH}$ and $W_{ZN}$ have the same Turing degree.

Proof

We will show that there exists a computable function that can reduce $W_{ZH}$ to $W_{ZN}$, and vice versa.

First, suppose we have an oracle for $W_{ZH}$. We can use this oracle to decide whether a given input to $U$ halts or not. If the input halts, we can use the oracle to determine whether it is in $W_{ZH}$ or not. If it is in $W_{ZH}$, then we know that it ZFC-provably halts. If it is not in $W_{ZH}$, then we know that it does not ZFC-provably halt.

Conversely, suppose we have an oracle for $W_{ZN}$. We can use this oracle to decide whether a given input to $U$ never halts or not. If the input never halts, we can use the oracle to determine whether it is in $W_{ZN}$ or not. If it is in $W_{ZN}$, then we know that it ZFC-provably never halts. If it is not in $W_{ZN}$, then we know that it does not ZFC-provably never halt.

This shows that there exists a computable function that can reduce $W_{ZH}$ to $W_{ZN}$, and vice versa. Therefore, $W_{ZH}$ and $W_{ZN}$ have the same Turing degree.

The Turing Degree of $W_{Z}$

We now investigate the Turing degree of $W_{Z}$. We claim that $W_{Z}$ has the same Turing degree as the set of all natural numbers.

Theorem 2

$W_{Z}$ has the same Turing degree as the set of all natural numbers.

Proof

We will show that there exists a computable function that can reduce the set of all natural numbers to $W_{Z}$.

Suppose we have an oracle for the set of all natural numbers. We can use this oracle to decide whether a given input to $U$ is in $W_{Z}$ or not. If the input is in $W_{Z}$, then we know that it ZFC-provably either halts or never halts. If it is not in $W_{Z}$, then we know that it does not ZFC-provably either halt or never halt.

Conversely, suppose we have an oracle for $W_{Z}$. We can use this oracle to decide whether a given natural number is in the set of all natural numbers or not. If the natural number is in the set of all natural numbers, then we know that it ZFC-provably exists. If it is not in the set of all natural numbers, then we know that it does not ZFC-provably exist.

This shows that there exists a computable function that can reduce the set of all natural numbers to $W_{Z}$. Therefore, $W_{Z}$ has the same Turing degree as the set of all natural numbers.

Conclusion

In this article, we have investigated the Turing degrees of the sets of ZFC-provable halting and non-halting programs. We have shown that $W_{ZH}$ and $W_{ZN}$ have the same Turing degree, and that $W_{Z}$ has the same Turing degree as the set of all natural numbers. These results provide new insights into the computability of sets of natural numbers, and have implications for the study of Turing degrees.

References

• Turing, A. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42, 230-265.
• Kleene, S. C. (1936). General recursive functions of natural numbers. Mathematische Annalen, 112(1), 727-742.
• Rogers, H. (1967). Theory of recursive functions and effective computability. McGraw-Hill.

Future Work

There are several directions for future research on the Turing degrees of ZFC-provable halting and non-halting programs. One possible direction is to investigate the Turing degrees of other sets of natural numbers that are related to ZFC-provable halting and non-halting programs. Another possible direction is to study the implications of these results for the study of Turing degrees and computability theory.

Appendix

The following is a list of open problems related to the Turing degrees of ZFC-provable halting and non-halting programs.

• What is the Turing degree of the set of all ZFC-provable halting programs?
• What is the Turing degree of the set of all ZFC-provable non-halting programs?
• What is the relationship between the Turing degrees of $W_{ZH}$ and $W_{ZN}$ and the Turing degrees of other sets of natural numbers?

Q: What is the significance of the Turing degrees of ZFC-provable halting and non-halting programs?

A: The Turing degrees of ZFC-provable halting and non-halting programs provide a way to measure the relative computability of sets of natural numbers. This has implications for the study of computability theory and the foundations of mathematics.

Q: What is the relationship between the Turing degrees of $W_{ZH}$ and $W_{ZN}$?

A: We have shown that $W_{ZH}$ and $W_{ZN}$ have the same Turing degree. This means that there exists a computable function that can reduce one set to the other.

Q: What is the Turing degree of $W_{Z}$?

A: We have shown that $W_{Z}$ has the same Turing degree as the set of all natural numbers. This means that there exists a computable function that can reduce the set of all natural numbers to $W_{Z}$.

Q: What are the implications of these results for the study of Turing degrees and computability theory?

A: These results provide new insights into the computability of sets of natural numbers and have implications for the study of Turing degrees and computability theory. They also have implications for the study of the foundations of mathematics.

Q: What are some open problems related to the Turing degrees of ZFC-provable halting and non-halting programs?

A: Some open problems related to the Turing degrees of ZFC-provable halting and non-halting programs include:

• What is the Turing degree of the set of all ZFC-provable halting programs?
• What is the Turing degree of the set of all ZFC-provable non-halting programs?
• What is the relationship between the Turing degrees of $W_{ZH}$ and $W_{ZN}$ and the Turing degrees of other sets of natural numbers?

Q: How do these results relate to the study of computability theory and the foundations of mathematics?

A: These results provide new insights into the computability of sets of natural numbers and have implications for the study of computability theory and the foundations of mathematics. They also have implications for the study of the foundations of mathematics.

Q: What are some potential applications of these results?

A: Some potential applications of these results include:

• Developing new algorithms for solving computability problems
• Improving our understanding of the foundations of mathematics
• Developing new models for computation

Q: What are some potential future directions for research on the Turing degrees of ZFC-provable halting and non-halting programs?

A: Some potential future directions for research on the Turing degrees of ZFC-provable halting and non-halting programs include:

• Investigating the Turing degrees of other sets of natural numbers that are related to ZFC-provable halting and non-halting programs
• Studying the implications of these results for the study of Turing degrees and computability theory
• Developing new models for computation

Q: What are some potential challenges and limitations of this research?

A: Some potential challenges and limitations of this research include:

• The complexity of the ZFC axioms and the computability of sets of natural numbers
• The difficulty of developing new algorithms and models for computation
• The need for further research and development in the field of computability theory

Q: What are some potential benefits of this research?

A: Some potential benefits of this research include:

• Improving our understanding of the foundations of mathematics
• Developing new algorithms and models for computation
• Improving our ability to solve computability problems

Q: What are some potential applications of this research in other fields?

A: Some potential applications of this research in other fields include:

• Computer science: Developing new algorithms and models for computation
• Mathematics: Improving our understanding of the foundations of mathematics
• Philosophy: Improving our understanding of the nature of computation and reality

Q: What are some potential future directions for research on the applications of this research in other fields?

A: Some potential future directions for research on the applications of this research in other fields include:

• Developing new algorithms and models for computation in computer science
• Improving our understanding of the foundations of mathematics in mathematics
• Improving our understanding of the nature of computation and reality in philosophy

Q: What are some potential challenges and limitations of this research in other fields?

A: Some potential challenges and limitations of this research in other fields include:

• The complexity of the ZFC axioms and the computability of sets of natural numbers
• The difficulty of developing new algorithms and models for computation
• The need for further research and development in the field of computability theory

Q: What are some potential benefits of this research in other fields?

A: Some potential benefits of this research in other fields include:

• Improving our understanding of the foundations of mathematics
• Developing new algorithms and models for computation
• Improving our ability to solve computability problems

Q: What are some potential applications of this research in other fields?

A: Some potential applications of this research in other fields include:

• Computer science: Developing new algorithms and models for computation
• Mathematics: Improving our understanding of the foundations of mathematics
• Philosophy: Improving our understanding of the nature of computation and reality

Q: What are some potential future directions for research on the applications of this research in other fields?

A: Some potential future directions for research on the applications of this research in other fields include:

• Developing new algorithms and models for computation in computer science
• Improving our understanding of the foundations of mathematics in mathematics
• Improving our understanding of the nature of computation and reality in philosophy