Proving the Existence of the Real Number Set in ZFC: A Step-by-Step Guide

Understanding the rigorous construction of the real number set, (mathbb{R}), based on the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) is fundamental to modern mathematics. This process not only solidifies the theoretical underpinnings of real analysis but also showcases the power and elegance of set theory. In this guide, we will outline the steps involved in constructing (mathbb{R}) from the set of natural numbers, (mathbb{N}), through the sets of integers, (mathbb{Z}), and rational numbers, (mathbb{Q}).

Step 1: Defining the Natural Numbers

The natural numbers (mathbb{N}) can be constructed using the Zermelo-Fraenkel axioms, typically starting with the empty set (emptyset) as (0). Subsequent natural numbers are defined as the set of all preceding natural numbers. Formally, this can be expressed as:

Step 2: Defining the Integers

The integers (mathbb{Z}) can be constructed as equivalence classes of ordered pairs of natural numbers ((a, b)), where each pair ((a, b)) represents the integer (a - b). Formally, this is defined as:

(mathbb{Z} {(a, b) mid (a, b) in mathbb{N} times mathbb{N}} / sim),

where ((a, b) sim (c, d)) if and only if (a d b c). This construction ensures that (mathbb{Z}) is a well-defined set within ZFC.

Step 3: Defining the Rationals

The rational numbers (mathbb{Q}) can be constructed as equivalence classes of ordered pairs of integers ((p, q)) where (q eq 0). Two pairs ((p_1, q_1)) and ((p_2, q_2)) are equivalent if:

(p_1 q_2 p_2 q_1).

Thus, we have:

(mathbb{Q} {(p, q) mid p in mathbb{Z}, q in mathbb{N} setminus {0}} / sim)

This construction is valid within ZFC, ensuring that the rationals are well-defined.

Step 4: Defining the Real Numbers

The real numbers (mathbb{R}) can be defined using Cauchy sequences of rational numbers. A Cauchy sequence ((a_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), we have:

(|a_n - a_m|

The set of all Cauchy sequences can be denoted as (mathbb{Q}^{mathbb{N}}), and we can define a relation on this set to identify equivalence classes of Cauchy sequences that converge to the same limit.

Step 5: Establishing the Real Numbers as a Set

Using the ZFC axioms, particularly the Axiom of Union, Axiom of Power Set, and Axiom of Replacement, we can show that the set of equivalence classes of Cauchy sequences forms a set, which we denote as (mathbb{R}).

Conclusion

By constructing (mathbb{N}), (mathbb{Z}), (mathbb{Q}), and then (mathbb{R}) using the axioms of ZFC, we establish the existence of the real numbers as a set. Each step relies on the foundational properties of sets and operations defined within ZFC, ensuring that (mathbb{R}) is well-defined in the context of set theory.