Peano Axioms For The Integers Z \mathbb{Z} Z

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Introduction

The Peano axioms are a set of fundamental principles that define the properties of the natural numbers. These axioms were first introduced by the Italian mathematician Giuseppe Peano in the late 19th century and have since become a cornerstone of modern mathematics. In this article, we will explore the possibility of defining the integers, including both positive and negative numbers, using a similar set of axioms.

Defining the Integers

For convenience, let's define the set of integers as $\mathbb{Z} = \{ ..., -3, -2, -1, 0, 1, 2, 3, ... \}$. Our goal is to define the integers using a set of axioms that are similar to the Peano axioms for the natural numbers.

Peano Axioms for the Natural Numbers

Before we proceed, let's briefly review the Peano axioms for the natural numbers:

• Peano Axiom 1: There exists a natural number, denoted by 0, such that for all natural numbers n, n ≠ 0.
• Peano Axiom 2: For every natural number n, there exists a unique natural number, denoted by n+, such that n+ ≠ n.
• Peano Axiom 3: For all natural numbers n, if n+ ≠ 0, then n+ is a natural number.
• Peano Axiom 4: For all natural numbers n, if n+ = m+, then n = m.
• Peano Axiom 5: There is no natural number n such that n+ = 0.

Extending the Peano Axioms to the Integers

To define the integers using a similar set of axioms, we need to introduce a new operation, denoted by -, which will allow us to "negate" a natural number. We can then define the integers as the set of all natural numbers and their negatives.

Axiom 1: Existence of 0

There exists an integer, denoted by 0, such that for all integers n, n ≠ 0.

Axiom 2: Existence of Negation

For every natural number n, there exists a unique integer, denoted by -n, such that -n ≠ n.

Axiom 3: Properties of Negation

For all natural numbers n, if -n ≠ 0, then -n is an integer. Furthermore, if n ≠ 0, then -(-n) = n.

Axiom 4: Commutativity of Addition

For all integers n and m, if n ≠ 0 and m ≠ 0, then n + m = m + n.

Axiom 5: Associativity of Addition

For all integers n, m, and p, if n ≠ 0 and m ≠ 0 and p ≠ 0, then (n + m) + p = n + (m + p).

Axiom 6: Existence of Additive Identity

There exists an integer, denoted by 0, such that for all integers n, n + = n.

Axiom 7: Existence of Additive Inverse

For every integer n, there exists a unique integer, denoted by -n, such that n + (-n) = 0.

Axiom 8: Trichotomy Law

For all integers n and m, exactly one of the following holds: n < m, n = m, or n > m.

Axiom 9: Transitivity of Less Than

For all integers n, m, and p, if n < m and m < p, then n < p.

Axiom 10: Total Ordering

For all integers n and m, if n ≠ m, then either n < m or m < n.

Conclusion

In this article, we have explored the possibility of defining the integers using a set of axioms that are similar to the Peano axioms for the natural numbers. We have introduced a new operation, denoted by -, which allows us to "negate" a natural number, and defined the integers as the set of all natural numbers and their negatives. The resulting axioms provide a foundation for the properties of the integers, including the existence of 0, the existence of negation, the properties of negation, the commutativity and associativity of addition, the existence of additive identity and inverse, the trichotomy law, the transitivity of less than, and the total ordering.

References

• Peano, G. (1889). Arithmetices principia, nova methodo exposita. Bocca.
• Russell, B. (1901). Principles of Mathematics. Cambridge University Press.
• Whitehead, A. N., & Russell, B. (1910). Principia Mathematica. Cambridge University Press.

Further Reading

• Introduction to Mathematical Logic by Elliott Mendelson
• A Course in Mathematical Logic by John L. Kelley
• Set Theory and Its Philosophy by Michael Potter
  Peano Axioms for the Integers $\mathbb{Z}$: Q&A =====================================================

Introduction

In our previous article, we explored the possibility of defining the integers using a set of axioms that are similar to the Peano axioms for the natural numbers. In this article, we will answer some of the most frequently asked questions about the Peano axioms for the integers.

Q: What are the Peano axioms for the integers?

A: The Peano axioms for the integers are a set of axioms that define the properties of the integers. They include the existence of 0, the existence of negation, the properties of negation, the commutativity and associativity of addition, the existence of additive identity and inverse, the trichotomy law, the transitivity of less than, and the total ordering.

Q: Why are the Peano axioms important?

A: The Peano axioms are important because they provide a foundation for the properties of the integers. They are used to prove many theorems in mathematics, including theorems about arithmetic, algebra, and geometry.

Q: What is the difference between the Peano axioms for the natural numbers and the Peano axioms for the integers?

A: The main difference between the Peano axioms for the natural numbers and the Peano axioms for the integers is the introduction of a new operation, denoted by -, which allows us to "negate" a natural number. This operation is not present in the Peano axioms for the natural numbers.

Q: Can the Peano axioms for the integers be used to prove the existence of negative numbers?

A: Yes, the Peano axioms for the integers can be used to prove the existence of negative numbers. The existence of negative numbers is a direct consequence of the existence of negation, which is one of the Peano axioms for the integers.

Q: Are the Peano axioms for the integers a complete set of axioms?

A: The Peano axioms for the integers are a complete set of axioms in the sense that they are sufficient to prove all the theorems about the integers. However, they are not a complete set of axioms in the sense that they do not provide a complete description of the integers.

Q: Can the Peano axioms for the integers be used to prove the existence of fractions?

A: No, the Peano axioms for the integers cannot be used to prove the existence of fractions. The Peano axioms for the integers are only sufficient to prove the existence of integers, and not fractions.

Q: Are the Peano axioms for the integers a set of axioms that can be used to prove theorems about real numbers?

A: No, the Peano axioms for the integers are not a set of axioms that can be used to prove theorems about real numbers. The Peano axioms for the integers are only sufficient to prove the existence of integers, and not real numbers.

Q: Can the Peano axioms for the integers be used to prove the existence of complex numbers?

A: No, the Peano axioms for the integers are not a set of axioms that can be used to prove the existence of complex numbers. The Peano axioms for the integers are only sufficient to prove the existence of integers, and not complex numbers.

Conclusion

In this article, we have answered some of the most frequently asked questions about the Peano axioms for the integers. We have discussed the importance of the Peano axioms, the difference between the Peano axioms for the natural numbers and the Peano axioms for the integers, and the limitations of the Peano axioms for the integers.

References

• Peano, G. (1889). Arithmetices principia, nova methodo exposita. Bocca.
• Russell, B. (1901). Principles of Mathematics. Cambridge University Press.
• Whitehead, A. N., & Russell, B. (1910). Principia Mathematica. Cambridge University Press.

Further Reading

• Introduction to Mathematical Logic by Elliott Mendelson
• A Course in Mathematical Logic by John L. Kelley
• Set Theory and Its Philosophy by Michael Potter