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nLab > Latest Changes: Dedekind cut

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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 RR vs UU 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∈Ub \in U, we have ¬¬(b∈U)\neg\neg(b \in U). Since ¬¬(b∈L)\neg\neg(b \in L) and ¬¬\neg\neg distributes over ∧\wedge, we have ¬¬(b∈L∧b∈R)\neg\neg(b \in L \wedge b \in R). Since b∈L∧b∈R⇒⊥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 ℚ +→P(ℚ)\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.

  4. Added a short remark that equality of Dedekind cuts is ¬¬\neg\neg-stable.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 17th 2018

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

    diff, v20, current

    • CommentRowNumber6.
    • CommentAuthorGuest
    • CommentTimeApr 14th 2022

    When editing is available again on the nLab, I think it would be a good idea to talk about the definition of Dedekind cuts in terms of σ\sigma-frames Σ\Sigma as the set of open truth values, such as Sierpinski space 𝕊\mathbb{S}, and pairs of open subspaces (L,U):(ℚ→Σ)×(ℚ→Σ)(L, U):(\mathbb{Q} \to \Sigma) \times (\mathbb{Q} \to \Sigma), as is common in predicative constructive mathematics and formal topology.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2023

    polished up the existing reference items

    and added these pointers:

    diff, v33, current

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