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