# Equicardinality of $\mathbb{R}$ and $\mathbb{R}^2$ via interleaving decimal expansions

As Fernando Q. Gouvêa notes in his paper, Was Cantor Surprised? (Amer. Math. Monthly 118 (March 2011), 198–209) Cantor initially tried to prove that $$(0,1]$$ and $$(0,1] \times (0,1]$$ have the same cardinality by representing each real number in $$(0,1]$$ by its unique non-terminating decimal expansion, and then interleaving the decimal expansions. Dedekind, however, pointed out a gap in this argument: numbers such as $$0.12010101010\ldots$$ do not correspond to any ordered pair of real numbers, because $$0.100000\ldots$$ is a terminating decimal expansion and hence not allowed.

There is simple but rather ingenious way to repair Cantor's argument. Instead of interleaving the decimal expansions one digit at a time, one breaks up the decimal expansion into "blocks," where a "block" denotes a maximal sequence of 0's followed by a nonzero digit, and then interleaves the blocks. More details may be found here, for example.

Who first came up with the repaired interleaving argument?

I have long thought that the answer was Dedekind himself, but I have not been able to back this belief up with a reference. In a comment on MathOverflow, Benjamin Dickman points out that the argument appears in Raymond L. Wilder's Introduction to the Foundations of Mathematics (Wiley, 1952), on page 89, but surely the argument is older than that.

Joseph Dauben's 1983 Scientific American article, Georg Cantor and the Origins of Transfinite Set Theory, gives the impression that the modified interleaving argument is due to Cantor himself. On page 126, Dauben refers to Cantor's 1877 correspondence with Dedekind, and directs the reader to page 129, where Dauben has a diagram of the modified interleaving argument. However, in his 1878 article in Crelle's journal, Ein Beitrag zur Mannigfaltigkeitslehre, Cantor uses continued fractions rather than decimal expansions, and exhibits, via interleaving, a bijection between $$I$$ and $$I^n$$, where $$I$$ is the set of irrational numbers in $$(0,1]$$. Perhaps the modified argument using decimal expansions appears elsewhere in Cantor's writings?