Construction of Real Numbers Using Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZF(C))

The foundation of real numbers in mathematical set theory, particularly within the framework of Zermelo-Fraenkel set theory with the Axiom of Choice (ZF(C)), is a profound and intricate process. This construction starts from basic set axioms and gradually builds to the complex structure of real numbers, ensuring a rigorous mathematical foundation.

1. Natural Numbers: The Building Block

The natural numbers, denoted by (mathbb{N}), are the most fundamental set in the construction of the real numbers. They can be constructed from the empty set using the successor function, which is a key concept in set theory. Specifically:

0 is defined as the empty set: (emptyset) 1 is defined as ({0} {emptyset}) 2 is defined as ({0, 1} {emptyset, {emptyset}}) In general, the natural number (n) is defined as the set of all natural numbers less than (n): (n {0, 1, 2, ldots, n-1})

2. Integers: Expanding the Number System

The integers, denoted by (mathbb{Z}), are constructed from the natural numbers by considering equivalence classes of ordered pairs of natural numbers. This construction allows for the introduction of negative values. Specifically, an integer can be represented as an equivalence class of pairs ((a, b)) where (a, b in mathbb{N}) and ((a_1, b_1) sim (a_2, b_2)) if (a_1 b_2 a_2 b_1).

3. Rational Numbers: Fractions and Equivalence

The rational numbers, denoted by (mathbb{Q}), provide a natural extension of the integers to include fractions. These are constructed as equivalence classes of pairs of integers ((p, q)) where (p in mathbb{Z}) and (q in mathbb{N} setminus {0}). Two pairs ((p_1, q_1)) and ((p_2, q_2)) are equivalent if (p_1 q_2 p_2 q_1). This construction guarantees that each rational number can be uniquely represented in its simplest form.

4. Real Numbers: Advanced Construction Methods

The real numbers, denoted by (mathbb{R}), can be constructed using several methods. Two common methods are the construction of Cauchy sequences and Dedekind cuts, which provide rigorous foundations for the real number system.

4.1 Cauchy Sequences

One method involves defining the real numbers as equivalence classes of Cauchy sequences of rational numbers. A Cauchy sequence ((x_n)) in (mathbb{Q}) is a sequence such that for every (epsilon > 0), there exists an (N) such that for all (m, n geq N), (|x_n - x_m|

4.2 Dedekind Cuts

An alternative method is through Dedekind cuts, which involves partitioning the rational numbers into two non-empty subsets (A) and (B) such that every element of (A) is less than every element of (B) and (A) contains no greatest element. Each cut corresponds to a unique real number.

Conclusion

The rigorous construction of the real numbers using Zermelo-Fraenkel set theory with the Axiom of Choice (ZF(C)) provides a solid foundation for analysis and ensures the completeness of the real number system. This process is crucial for establishing the properties of real numbers and forms the basis for much of modern mathematics.