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Statement

A set of real numbers is rationally independent when it is linearly independent over the rationals. The continuum hypothesis is $2^{\aleph_0}=\aleph_1$.

Theorem 2 (p. 459, quoted). "The continuum hypothesis is equivalent to the following proposition:

(P) The set of all real numbers can be decomposed into a countable number of subsets, each consisting only of rationally independent numbers."

Proof pointer

Pp. 459--460. CH implies (P): take a Hamel basis indexed by the countable ordinals; split the nonzero reals by the finite sequence of nonzero rational coefficients in their expansion, and each of these countably many classes by the position of the expansion's largest basis index, which leaves countable pieces; enumerating each piece and taking one element from each piece at each position gives countably many sets in which distinct elements have distinct largest basis indices. In such a set a vanishing integer combination would contain the basis element of the largest index exactly once, which is impossible. (P) implies CH: by König's theorem one of the sets, $M_1$, has the power of the continuum. On the vertex set $M_1$, put the segment from $x$ to $y>x$ into the $n$-th graph when $y-x\in M_n$. A closed polygon in one graph would give a vanishing signed sum of the differences $\lvert x_i-x_{i+1}\rvert$, which lie in $M_n$ and, as the vertices lie in $M_1$, are all different, contradicting the independence of $M_n$. So the complete graph on $2^{\aleph_0}$ vertices is a union of countably many trees, and Theorem 1 gives $2^{\aleph_0}\leqq\aleph_1$.

Read depth

Claims checked: Theorem 2 was read clause by clause on the page images of the print, and its proof (pp. 459--460) was followed. Nothing here is independently reviewed.

Dependencies

Theorem 1 of the same paper. External inputs named by the paper: the existence of a Hamel basis and König's theorem (cited from Sierpiński, Hypothèse du continu, p. 6).

Source. P. Erdős and S. Kakutani, On non-denumerable graphs, Bull. Amer. Math. Soc. 49 (1943), 457--461, doi:10.1090/S0002-9904-1943-07954-2; the edition read is named on the source card.

Bears on

• Problem 1127: the paper states no result about distances. Under the continuum hypothesis Theorem 2 splits the real line into countably many rationally independent sets, and in a rationally independent set two different pairs of points never have the same distance, a fact the proof on p. 460 uses for $M_1$. So the theorem gives a yes answer for $n=1$ under the continuum hypothesis. Its converse half concerns rationally independent sets only and does not show that the continuum hypothesis is needed for distinct distances.