Part II, Chapter I — Analytical Summary

Countability of $\overline{\mathbb{Q}} \cap \mathbb{R}$ vs. Uncountability of $\mathbb{R}$ (§§ 1–3)

Logical Structure of Part II’s Opening Argument

Before proving that the specific constants $e$ and $\pi$ are transcendental (Chapters III–IV), Klein opens Part II with Georg Cantor's set-theoretic demonstration (1874, Halle 1891) that transcendental numbers not only exist, but overwhelm the algebraic numbers in every sub-interval of the line:

  1. Dense $\neq$ Continuous (§ 1): Rational numbers are dense (überall dicht) on $\mathbb{R}$, and real algebraic numbers $\overline{\mathbb{Q}} \cap \mathbb{R}$ (roots of $a_0 \omega^n + \dots + a_n = 0$, $a_i \in \mathbb{Z}$) are denser still—yet neither exhausts the continuum.

  2. Enumeration by Algebraic Height (§ 2): Assigning each irreducible coprime integer polynomial ($a_0 > 0$) its height $N = n - 1 + \sum_{i=0}^n |a_i|$ partitions $\overline{\mathbb{Q}} \cap \mathbb{R}$ into finite layers ($\varphi(1) = 1$, $\varphi(2) = 2$, $\varphi(3) = 4$, $\varphi(4) = 12, \dots$), placing all real algebraic numbers in bijection (gleiche Mächtigkeit) with $\mathbb{Z}^+$.

  3. Diagonal Avoidance (§ 3): Fixing any finite decimal prefix (hence any sub-interval) and choosing each subsequent digit $\neq 9$ and $\neq$ the diagonal digit of $\omega_k$ produces $8^{\aleph_0}$ non-algebraic reals in that interval alone.