Dedekind Cuts, Moving Markers, and the Uncountability of ℝ

Summary

We give short proofs that $ℝ$ is uncountable directly from the definition of $ℝ$ as the set of Dedekind cuts of $\mathbb{Q}$.

MSC:

• 03E99

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Another common way to define $ℝ$ is as a set of equivalence classes of Cauchy sequences of rationals. The paper by Wenner [5] gives a natural proof that this set of equivalence classes is uncountable.

2 In fact, some real numbers have two such representations, for example $0.500000\dots =0.499999\dots$; the choice of 7 and 8 in the construction of y avoids any problems associated with this non-uniqueness. Petkovšek [6] and Starbird and Starbird [7] have shown that this is unavoidable.

Additional information

Notes on contributors

David A. Ross

David A. Ross (MR Author ID: 232420) received his Ph.D. from the University of Wisconsin, Madison, and is a professor of mathematics at the University of Hawai‘i at Mānoa. He is interested in most areas of mathematics, but has also published in philosophy, computer science, and zoology. His most enduring legacy will probably be the eponymous ‘droplet technique’ for reducing static in coffee grinding.