Part IV: The Number Hierarchy

The number systems form a nested hierarchy, each extending the previous to solve equations that were insoluble before. The hierarchy is not arbitrary: each extension is forced by specific algebraic requirements.

4.1 Natural Numbers ℕ

Objects: 0, 1, 2, 3, …

Peano Axioms:

1. 0 ∈ ℕ
2. Every natural number n has a unique successor S(n) ∈ ℕ
3. 0 is not a successor of any natural number
4. If S(m) = S(n), then m = n (successors are injective)
5. (Induction) If a property P holds for 0, and P(n) implies P(S(n)), then P holds for all natural numbers.

Set-theoretic construction: Define 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, … Each natural number n is represented as the set of all smaller natural numbers. The Axiom of Infinity guarantees this construction terminates in a set.

Arithmetic: Addition and multiplication are defined recursively from the successor function. The fundamental theorem of arithmetic: every natural number > 1 has a unique prime factorization.

What ℕ lacks: Subtraction is not always defined. x − y requires y ≤ x.

4.2 Integers ℤ

Construction: Pairs (a, b) of natural numbers, with (a,b) representing a − b, under the equivalence (a,b) ~ (c,d) ↔ a+d = b+c.

What ℤ adds: Additive inverses. Every integer has a negative. Subtraction is always defined.

Algebraic structure: (ℤ, +, ×) is an integral domain — a commutative ring with no zero divisors.

What ℤ lacks: Multiplicative inverses. 2/3 is not an integer. Division is only sometimes defined.

4.3 Rational Numbers ℚ

Construction: Pairs (p, q) with q ≠ 0, representing p/q, under (p,q) ~ (r,s) ↔ ps = qr.

What ℚ adds: Multiplicative inverses for all non-zero elements. Division is always defined (by non-zero elements).

Algebraic structure: (ℚ, +, ×) is a field.

What ℚ lacks: Completeness. The sequence 1, 1.4, 1.41, 1.414, … converges in a limiting sense to √2, but √2 ∉ ℚ (proved by Pythagoreans: if √2 = p/q in lowest terms, then p² = 2q², so p is even, say p = 2r, then 2r² = q², so q is even, contradicting lowest terms).

Density: ℚ is dense in ℝ — between any two distinct rationals there is another. But ℚ has "holes."

4.4 Real Numbers ℝ

Dedekind Cuts (1872): A real number is a cut — a partition (A, B) of ℚ into a left set A and right set B, with every element of A less than every element of B, A non-empty, B non-empty, and A having no greatest element. √2 corresponds to the cut A = {q ∈ ℚ : q ≤ 0 or q² < 2}, B = {q ∈ ℚ : q > 0 and q² ≥ 2}.

Cauchy Sequences (alternative): Equivalence classes of Cauchy sequences of rationals — sequences where |a_n − a_m| → 0 as n,m → ∞ — under the equivalence of sequences that converge to the same limit.

Completeness: The completeness axiom (or least upper bound property): every non-empty subset of ℝ that is bounded above has a supremum (least upper bound) in ℝ. This is what Dedekind cuts enforce.

Key properties:

• ℝ is an ordered field
• ℝ is complete (no holes)
• ℝ is uncountable: |ℝ| = 2^ℵ₀ > ℵ₀ = |ℕ| (Cantor's diagonal argument)
• ℝ is connected: the only subsets that are both open and closed are ∅ and ℝ
• The Archimedean property: for any x ∈ ℝ, there exists n ∈ ℕ with n > x

What ℝ lacks: A solution to x² + 1 = 0.

4.5 Complex Numbers ℂ

Construction: Pairs (a, b) of real numbers, with (a,b) representing a + bi, where i is a formal symbol with i² = −1.

Operations:

• (a + bi) + (c + di) = (a+c) + (b+d)i
• (a + bi)(c + di) = (ac − bd) + (ad + bc)i

Fundamental Theorem of Algebra: Every non-constant polynomial with complex coefficients has a root in ℂ. Equivalently, ℂ is algebraically closed.

Geometric interpretation: ℂ ≅ ℝ² as a plane. Multiplication by z = re^(iθ) is rotation by θ and scaling by r. This makes ℂ the natural setting for 2D geometry and Fourier analysis.

What ℂ loses: ℂ is not an ordered field. There is no consistent total ordering on ℂ compatible with its field operations.

4.6 The Division Algebras

Hamilton's Quaternions ℍ (1843):

• Dimension 4 over ℝ: {a + bi + cj + dk : a,b,c,d ∈ ℝ}
• Rules: i² = j² = k² = ijk = −1
• Non-commutative: ij = k but ji = −k
• Every non-zero quaternion has a multiplicative inverse
• Applications: 3D rotation (SU(2) ≅ unit quaternions), aerospace navigation, quantum mechanics

Octonions 𝕆 (Graves, Cayley, 1843):

• Dimension 8 over ℝ
• Non-associative: (xy)z ≠ x(yz) in general
• Alternative: (xx)y = x(xy) and x(yy) = (xy)y
• Applications: exceptional Lie groups, string theory, M-theory

Frobenius Theorem: The only finite-dimensional real division algebras are ℝ, ℂ, ℍ, 𝕆. The Cayley-Dickson construction beyond 𝕆 loses the division property. The hierarchy terminates.

4.7 The p-adic Numbers ℚ_p

Motivation: Completeness requires a notion of "closeness." The standard metric |x − y| is not the only option. For a prime p, define the p-adic absolute value |n|_p = p^(−v_p(n)) where v_p(n) is the p-adic valuation (highest power of p dividing n).

Ostrowski's Theorem: Every non-trivial absolute value on ℚ is equivalent either to the standard absolute value or to a p-adic absolute value for some prime p.

ℚ_p: The completion of ℚ with respect to the p-adic metric. Its elements are formal Laurent series in p: ∑ aᵢpⁱ for i ≥ N, with 0 ≤ aᵢ < p.

Properties:

• ℚ_p is a complete, locally compact, totally disconnected field
• The p-adic integers ℤ_p = {x ∈ ℚ_p : |x|_p ≤ 1} are compact
• The adeles 𝔸 = ℝ × ∏_p ℚ_p are fundamental in number theory

4.8 Surreal Numbers

Conway's construction (1970s): Surreal numbers are defined by a simple recursive rule: a surreal number is a pair {L | R} of sets of previously constructed surreal numbers, with no element of L greater than or equal to any element of R.

What surreals contain:

• All real numbers
• All infinite ordinals (ω, ω², …)
• All infinitesimals (1/ω, 1/ω², …)
• Numbers like ω − 1, √ω, ω^(1/ω)

**The surreals form a proper class, not a set — they are "too large" to be a set in ZFC.

Structure: (No, ≤, +, ×) is a totally ordered field containing ℝ as a subfield and Ord (the ordinals) as a sub-semiring.

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