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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.
I have added a sentence mentioning forcing to the Idea-section at double negation,
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.
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.
Yes, it seems that statement had been there since Day 1. I suspect Toby was thinking along roughly the following lines: a topological space $X$ in which every open is regular open is (classically) discrete. If you allow the luxury of $T_1$ spaces, and if $x$ is any non-isolated point, then its set-theoretic complement $U$ has $U \cup \{x\} = X$ as its closure, so $\neg \neg U = X$. So maybe with a sufficiently generous interpretation of “classical”, the statement is defensible – but I agree the statement is confusing as it stands.
Maybe it should be “If $L$ is spatial, then $L=L_{\neg\neg}$ if and only if …”?
Yes, spatiality is certainly necessary here.
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