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Statement

Theorem (p. 461, unnumbered). The continuum has power $\aleph_{x+1}$ if and only if both of the following hold: the set $R$ of real numbers is the union of $\aleph_x$ sets, each consisting of rationally independent numbers; and $R$ is not the union of fewer than $\aleph_x$ such sets.

Proof pointer

P. 461: the paper says only that the proof is the same as that of Theorem 2. In that proof the role of Theorem 1 would fall to the $\aleph_x$ remark stated without proof on p. 459 (see Theorem 1); the paper does not say this.

Read depth

Claims checked: the statement was read clause by clause on the page image of the print. The paper gives no separate proof. Nothing here is independently reviewed.

Dependencies

Theorem 2 and the p. 459 remark on Theorem 1.

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

No Erdős problem directly.