Structure of real numbers in formal mathematics

Formal Construction of Real Numbers from Foundational Structures

Real numbers ℝ are not introduced intuitively in formal mathematics, but through a step-by-step extension of the natural numbers. Starting with Peano arithmetic—where numbers are built from 0 and the successor function S—the system evolves through integers ℤ, rationals ℚ, and finally real numbers ℝ. This approach eliminates definitional gaps and clarifies 'oddities' of infinity, such as the non-commutativity of ω + 1 ≠ 1 + ω.

Natural numbers are defined recursively:

1 := S(0)
2 := S(S(0))
...

Addition is defined by rules:

a + 0 := a
a + S(b) := S(a + b)

This allows computation of 2 + 2 via substitution: 2 + 2 = S(S(2 + 0)) = S(S(2)) = 4. Multiplication follows similarly:

a · 0 := 0
a · S(b) := a · b + a

Example: 5 · 3 = 5 · 2 + 5 = 5 · 1 + 5 + 5 = 15.

Integers as Equivalence Classes

Integers ℤ extend ℕ to include negative values using ordered pairs of natural numbers: i = (a, b), where +5 = (5, 0) and -5 = (0, 5). In set theory, a pair is encoded via Kuratowski: (a, b) := {{a}, {a, b}}.

Addition: (a, b) + (c, d) := (a + c, b + d).

Multiplication: (a, b) · (c, d) := (a·c + b·d, a·d + b·c).

Pairs are equivalent if a + d = c + b. Each integer is an equivalence class of such pairs. The cardinality |ℤ| = |ℕ|, since a bijection exists: even numbers for positives, odd for negatives.

Rational Numbers and Their Cardinality

Rationals ℚ are pairs of integers (a, b) with b ≠ 0, equivalent under a·d = c·b. Multiplication: (a, b) · (c, d) := (a·c, b·d). Addition: (a, b) + (c, d) := (a·d + c·b, b·d).

|ℚ| = |ℕ|: positive rationals are arranged in a grid x/y, traversed diagonally (zigzag) to yield a bijection with ℕ.

• Diagonal 1: 1/1
• Diagonal 2: 1/2, 2/1
• Diagonal 3: 3/1, 2/2, 1/3

Ignoring equivalent fractions (e.g., 2/2 = 1/1), this traversal covers all ℚ⁺.

The Challenge of Real Numbers

Real numbers ℝ cannot be reduced to pairs or equivalence classes like ℚ. Dedekind introduced ℝ as cuts of rationals: a partition of ℚ into sets A and B such that ∀a∈A < ∀b∈B, A has no maximum, and B has no minimum.

For example, √2 corresponds to {q ∈ ℚ | q² < 2} and {q ∈ ℚ | q² ≥ 2}.

Each cut uniquely defines a real number. Arithmetic on cuts:

• Addition: A₁ ∪ A₂ (with rational correction).
• Multiplication: via squares.

Cardinality and the Continuum

|ℝ| > |ℚ|: Cantor’s diagonal argument. Assume a bijection f: ℕ → (0,1). Construct r where r_i ≠ f(i)_i — contradiction.

r = 0.r₁r₂r₃...
where r_i ≠ (f(i))_i

Thus, the continuum 2^ℵ₀ > ℵ₀. Ordinals and cardinals diverge: ω + 1 > ω, yet |ω| = |ω + 1|.

Key Takeaways

• Natural numbers: recursive construction from 0 and S, |ℕ| = ℵ₀.
• Integers and rationals: pairs with equivalence relations, same cardinality ℵ₀.
• Reals: Dedekind cuts or Cauchy sequences, |ℝ| = 2^ℵ₀.
• Infinities are non-commutative: ω + 1 ≠ 1 + ω.

— Editorial Team