> No, that wasn't my reasoning. Right or wrong, my position in Lecture 6 was this: I wasn't assuming the posets in question have all infs and sup, I was *claiming* that they *must* have the infs and sups in question, given my assumptions.
> ...
> That said, if someone gave me a puzzle whose answer was \\(a\\), and I said the answer was \\( \bigvee \\{b \in A: \; a \ge b \\} \\), we'd have to say my answer wasn't the best available, because I failed to simplify it as much as possible.


I was thinking like this: the *most general* way to think about Galois connections is on preorders. But this is annoying because they don't obey the anti-symmetry rule. They don't have infima and suprema which are natural.

However, I'm arguing there's a place a we can go: *The Dedekind-MacNeille Completion Functor*. If we embed our preorder up there, now we've got a real partial order like we've always wanted. We've even got sets which is nice. And we've got suprema and infima. And, when I can get around to it, I think I can prove a transfer theorem for adjunctions and fixed points.

(Transfer is my idea, but I got the idea of using it to transform preorders from [Erné (1991)](https://link.springer.com/article/10.1007/BF00383401).)

Here's a parallel: the *textbook* way to think about derivatives in calculus is with the \\(\delta-\epsilon\\) formulation on a real closed Archimedean field. But this is annoying because there's a lot of quantifiers and those are hard. Also, we don't have infinitesimals or their reciprocals which are natural (for Euler and Leibniz, anyway). Even Archimedes found it natural to use infinitesimals and break the rules that are his namesake in his [lost palimpsest](https://en.wikipedia.org/wiki/Archimedes_Palimpsest). And we can have it all with the Robinson's ultraproduct construction, and we have the transfer theorem for first order propositions.

Now, I can see why maybe it's annoying. Nobody really uses nonstandard analysis for much because it's hard to motivate and ultraproducts are clumsy. But for some, it validates their intuition. And I say Dedekind-MacNeille completions do the same for preorders. But that's just my opinion.