Chapter II: The Compatibility and Mutual Independence of the Axioms — Summary¶
Structural & Mathematical Overview — Kapitel II: Die Widerspruchslosigkeit und gegenseitige Unabhängigkeit der Axiome
Architectural Summary of Chapter II (§§ 9–12)¶
Having erected the five axiom groups in Chapter I, Hilbert inaugurates model-theoretic metamathematics in Chapter II, establishing both compatibility (Widerspruchslosigkeit) and mutual independence (Unabhängigkeit) through four explicit analytic coordinate models:
§ 9 — Relative Consistency via the Pythagorean Field $\Omega$: Constructs the countable algebraic number domain $\Omega \subset \mathbb{R}$ generated from $1$ by finite applications of $+, -, \times, \div$ and the fifth operation $\sqrt{1 + \omega^2}$. Defining points as $(x, y) \in \Omega^2$, lines as homogeneous ratios $(u : v : w)$, order via algebraic inequalities, and displacements via rational/Pythagorean orthogonal rotations, Hilbert proves that every contradiction deducible from Axioms I–V would already appear in the arithmetic of $\Omega$.
§ 10 — Independence of the Parallel Axiom III: Proves first that existence of parallels follows from I, II, IV alone (via alternate angles, Euclid I.27), and then restricts the $\Omega$-space of § 9 to the interior of a fixed sphere equipped with projective Cayley–Klein collineations preserving the sphere, yielding a consistent Bolyai–Lobachevsky hyperbolic geometry in which I, II, IV, V hold while III fails.
§ 11 — Independence of Congruence Axiom IV 6 (SAS): Retains ordinary points, lines, planes, and angles, but redefines segment length between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ by the anisotropic norm
$$\sqrt{(x_1 - x_2 + y_1 - y_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}.$$
Linear parameter proportionality preserves segment additivity (IV 3), so I, II, III, IV 1–5, V all hold, yet right triangles $OAC$ and $OBC$ with $O(0,0), A(1,0), B(0,1), C(\tfrac{1}{2}, \tfrac{1}{2})$ share $\angle C$, $OC$, and $AC = BC = \tfrac{1}{\sqrt{2}}$ while $OA = 1 \neq \sqrt{2} = OB$, falsifying SAS (Theorem 10 and IV 6).
§ 12 — Independence of Archimedes’s Axiom V: Replaces numbers by the countable field $\Omega(t)$ of real algebraic functions of a parameter $t$ generated by $+, -, \times, \div$ and $\sqrt{1 + \omega^2}$, ordered by sign for sufficiently large $t > 0$. Since $n - t < 0$ for every positive integer $n$ as $t \to +\infty$, the segment $1$ laid off $n$ times never reaches $t$, establishing a consistent non-Archimedean geometry satisfying I–IV while violating V.