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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2017
    • (edited May 9th 2017)

    I have added to the References at double negation pointer to Andrej’s exposition:

    which is really good. I have also added this to double negation transformation, but clearly that entry needs some real references, too.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2017

    I have added a sentence mentioning forcing to the Idea-section at double negation,

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 1st 2017

    That is a nice post, but I have to take issue with

    classical logic is happy with lack of negative evidence.

    I would say that what classical logic is happy with is the impossibility of negative evidence, which is rather stronger than the present lack thereof.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 23rd 2019

    The article claims

    Classically, we have L=L¬¬ if and only if L is the discrete locale on some set S of points. In constructive mathematics, S must also have decidable equality.

    But any complete Boolean algebra is a frame for which the corresponding locale satisfies L=L_¬¬ because ¬¬=id. There are plenty of nonatomic complete Boolean algebras.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 23rd 2019

    Yes, it seems that statement had been there since Day 1. I suspect Toby was thinking along roughly the following lines: a topological space XX in which every open is regular open is (classically) discrete. If you allow the luxury of T 1T_1 spaces, and if xx is any non-isolated point, then its set-theoretic complement UU has U{x}=XU \cup \{x\} = X as its closure, so ¬¬U=X\neg \neg U = X. So maybe with a sufficiently generous interpretation of “classical”, the statement is defensible – but I agree the statement is confusing as it stands.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2019

    Maybe it should be “If LL is spatial, then L=L ¬¬L=L_{\neg\neg} if and only if …”?

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 23rd 2019

    Yes, spatiality is certainly necessary here.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJul 24th 2019

    Add spatiality condition to the characterization of L=L ¬¬L=L_{\neg\neg}.

    diff, v58, current