Talk:Intersection is Decreasing

It's hard to believe we don't have this result somewhere, by some name, but I couldn't find it. I'll be glad to redirect if we have it already.--Dfeuer (talk) 18:51, 27 February 2013 (UTC)

I think a link to the word decreasing should be on the page somewhere for navigational purposes. --Jshflynn (talk) 19:25, 27 February 2013 (UTC)

Maybe that's better, except for the fact that we likely don't yet actually define mappings between classes, or decreasing mappings between classes.... --Dfeuer (talk) 19:39, 27 February 2013 (UTC)

We could fix that by establishing that $\mathcal F$ and $\mathcal G$ are subsets of the power set of a certain set, I suppose. Intersections are altogether difficult once you start dealing with empty ones. --Dfeuer (talk) 19:42, 27 February 2013 (UTC)

Because the result is simple it may be better to add a section at the bottom with this extra detail where we justify the naming of the page. E.g. "We may interpret this result using Set Theory and Order Theory as set intersection being a decreasing mapping ..." "... Hence the name of the page."

That way neither accessibility nor detail is lost. --Jshflynn (talk) 20:04, 27 February 2013 (UTC)

How do you like the current version? Perhaps a note at the bottom could mention that in some set-theoretic contexts we may replace $U$, $\mathcal P(U)$, and $\mathcal P(\mathcal P(U))$ with $\mathcal U$, the universe of sets? --Dfeuer (talk) 20:21, 27 February 2013 (UTC)

That will currently clash too much with the definition of mapping. Current version is kind of fine (I won't bother with the minute differences from global style that reside in my subconsciousness) and I suggest it isn't cluttered more. — Lord_Farin (talk) 20:23, 27 February 2013 (UTC)

That's fine for now, although we have a lot of work ahead of us to deal with axiomatic set theories. Mostly I wanted this for a simple theorem on closure operators, but it seems likely to be useful for other things too, even if we ignore the bit about a decreasing mapping. --Dfeuer (talk) 20:33, 27 February 2013 (UTC)

't Would in my opinion (and I've said things to the same extent on other occasions as well) be best to introduce some new approach proper in the first place before throwing around bits and pieces that may bring up unfounded expectations or, even worse, confuse the uninitiated reader. — Lord_Farin (talk) 20:35, 27 February 2013 (UTC)

Can you be more concrete/specific? I don't really understand your point.--Dfeuer (talk) 20:52, 27 February 2013 (UTC)

Instead of referring to "certain set-theoretic contexts" I consider it better to wait until these are actually covered so as to avoid confusion. 't Was a bit unrelated perhaps but I felt like stating that somehow. If anything, please try to keep that approach in mind (it can be applied e.g. to manifold theory and the Takeuti/Zaring axiomatic set theory, to name two recent examples); it will help keep the site structurally sound. — Lord_Farin (talk) 20:59, 27 February 2013 (UTC)

BTW, I think some of these things are inherently confusing, because it's possible to hang the structure up from various points. E.g., either you can prove that an operator is a closure operator, from which you know intersections of its closed sets are closed, or you prove intersections of subsets of a set are in the set, from which you know they induce a closure operator. --Dfeuer (talk) 20:57, 27 February 2013 (UTC)

(One of) Our main aim(s) is to remove such confusion as best as we can, right? View it as a challenge :). — Lord_Farin (talk) 20:59, 27 February 2013 (UTC)