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1. Added to Dedekind cut a short remark on the $\neg\neg$-stability of membership in the lower resp. the upper set of a Dedekind cut.

• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeJan 27th 2014

Interesting! I didn't quite follow the last bit of your argument, so I rephrased it. (I also regularized the notation of $R$ vs $U$ and finagled a link to stable property.)

• CommentRowNumber3.
• CommentAuthorIngoBlechschmidt
• CommentTimeJan 27th 2014
• (edited Jan 27th 2014)

Thanks for catching the typo and streamlining the argument! (For the record, my reasoning was as follows: Since $b \in U$, we have $\neg\neg(b \in U)$. Since $\neg\neg(b \in L)$ and $\neg\neg$ distributes over $\wedge$, we have $\neg\neg(b \in L \wedge b \in R)$. Since $b \in L \wedge b \in R \Rightarrow \bot$ and $\neg\neg$ is monotone, we have $\neg\neg\bot$, so $\bot$.)

The almost-$\neg\neg$-stability can be helpful when proving the equivalence of Dedekind cuts with multi-valued Cauchy “sequences” (i.e. certain maps $\mathbb{Q}^+ \to P(\mathbb{Q})$). Depending on one’s line of thought, of course, one can otherwise get stuck when trying to show that equivalent multi-valued Cauchy sequences define the same Dedekind cut.

2. Added a short remark that equality of Dedekind cuts is $\neg\neg$-stable.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeJul 16th 2018

Added Project Gutenberg link for Beman’s English translation of Continuity and irrational numbers

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