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The pages MacNeille real number and MacNeille completion disagree about whether $\pm\infty$ are MacNeille real numbers. The former says the MacNeille reals are the MacNeille completion of the rationals and hence contain $\pm\infty$, whereas the latter says that the MacNeille reals are obtained by “dropping $\pm\infty$” from that completion. Which should it be?
The Elephant also contains a definition of “MacNeille real number” that is more general than a Dedekind cut, but doesn’t contain $\pm\infty$: instead of “locatedness” (if $a\lt b$ are rationals then $a\in L$ or $b\in U$) they satisfy “if $a\lt b$ and $a\notin L$ then $b\in U$” and “if $a\lt b$ and $b\notin U$ then $a\in L$”. Since $L$ and $U$ are both required to be inhabited, this definition also excludes $\pm\infty$. But it’s not clear to me how it’s related to the general definition of MacNeille completion. It would be nice to include a comparison at the page MacNeille real number.
No one ever responded to this. I’m inclined to say that the “MacNeille real numbers” should exclude $\pm\infty$, to match the Elephant and also the terminology for other kinds of real number; then the full MacNeille completion would be the “extended MacNeille real numbers”. If no one objects I will implement that terminology.
I also notice that Toby edited MacNeille real number last week to add a definition, but it’s not the same as the definition in the Elephant, which requires $L$ and $U$ to be open rather than closed (hence themselves a lower and an upper real, respectively). I can believe that they’re equivalent classically, but constructively it seems more doubtful, and probable that what we want is the open ones? (The Elephant’s definition is a pair $(L,U)$ of a lower and an upper real such that each is the interior of the complement of the other.)
Now I see, though, how the definition with closed sets is an instance of the MacNeille completion.
Yes, that's why I put it in. Then I decided that perhaps we should go with open so as to make the different kinds of real numbers interoperable, so that it's easier to say that a certain MacNeille real number ‘is’ a Dedekind real number, etc. (The one-sided real numbers use one-sided cuts, although it should be easy enough to add the other side to make them into MacNeille real numbers.) But I would need to check their equivalence, and then I didn't have time to keep working on it.
On the other issue, I like the progression from Dedekind to one-sided to MacNeille real number, that not only broaden constructively, but also broaden classically in their treatment of infinity. (One can always add ‘bounded’ or ‘extended’ to modify this.) In the case of the one-sided real numbers, it seems to be an important principle that they naturally include one kind of infinity but not the other; in the case of the Dedekind and MacNeille reals, this is not such a big deal. (But the one-sided real numbers do contradict your claim that ‘other kinds of real number’ are always bounded.)
But there is also something to be said for having a progression of two-sided, bounded real numbers, all classically equivalent: modulated Cauchy, Cauchy, Dedekind, MacNeille, Conway (possibly others).
But given what you've just written at the one-sided real number discussion, my conception that the MacNeille real numbers are a common generalization of the one-sided real numbers is just wrong. Indeed, if $\mathbb{R}_m = \mathbb{R}_l \cap \mathbb{R}_u$ within the surreal numbers, then it is indeed most natural to make all MacNeille numbers bounded.
Actually, I think that the closed and open definitions of MacNeille reals coincide even constructively. I added a proof to MacNeille real number, as well as some discussion of the relation to other kinds of real number, and changed the terminology to exclude infinities unless we say “extended”.
One thing that can be said precisely about the inclusion of infinities is that the lower reals (including $+\infty$ but not $-\infty$), the upper reals (including $-\infty$ but not $+\infty$), and the located reals (including neither $+\infty$ nor $-\infty$) are the models of a propositional geometric theory, and hence are the points of naturally defined locales. In all cases we can include $\pm\infty$ if we want and maintain this property, but we cannot exclude $+\infty$ from the lower reals or $-\infty$ from the upper reals geometrically. However, the MacNeille axiom already fails to be geometric, so this point of view doesn’t tell us anything about whether $\pm\infty$ should be included.
The only relationship I see is that they’re both defined using “complemented subsets”, i.e. pairs of subsets $(A,B)$ with $A\cap B = \emptyset$. That means in particular that both of them are naturally defined in linear constructive mathematics (which I didn’t know before about Cheng spaces, although our page on them does mention that complemented subsets are a Chu construction), so if we had a page about that then we could link both of them to it. But I don’t immediately see anything that could be said linking them together directly — do you?
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