Talk:Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest

Probably a proof along the characteristics that the intersection of transitive relations is transitive, well, shortly, like Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets can be made. Nice and intrinsic. --Lord_Farin 18:07, 6 April 2012 (EDT)

I don't follow you ... --prime mover 18:09, 6 April 2012 (EDT)

Usually, there is some trivial structure (like the power set, or Cartesian product in the present case) both encompassing the structure that is to be expanded and having the desired properties.

Mostly, when one speaks about the smallest ladidada, this is WRT subset relation, and that usually implies that ladidada is preserved under arbitrary intersection. So the proof for ref'd results almost literally carries over. --Lord_Farin 18:14, 6 April 2012 (EDT)

I confess that this-all is actually somewhat outside my comfort zone - I posted it up when I was doing some specific research on graph theory for a java project for my day-job, and the pages on WP of R- S- and T-closure had direct relevance and looked interesting enough to post up. Beyond what I needed for work (which Neal Stephenson coincidentally covered directly in his Anathem novel) I know too little to be able to continue down this route. Feel free to take it from here. --prime mover 03:33, 7 April 2012 (EDT)

I put it up, does it make more sense now? --Lord_Farin 11:08, 19 April 2012 (EDT)

Works for me.--prime mover 12:48, 19 April 2012 (EDT)

Refactor

Since proof 1 was really about constructing the transitive closure by countable unions, with some repetition of proof 2 on top, I moved the good parts to Recursive Construction of Transitive Closure. I'm not sure that's really the best way to refactor, but I think this at least makes more sense than what came before. Proof 2 is clearly the simplest way by far to prove this theorem. --Dfeuer (talk) 21:17, 13 December 2012 (UTC)

This article contains statements that are justified by handwavery. In particular: Prove the final statement of the above formally. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding precise reasons why such statements hold. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Handwaving}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

The two proofs currently here are essentially the same. There are different proofs at Recursive Construction of Transitive Closure. Someone needs to decide how to arrange these things more sensibly. --Dfeuer (talk) 19:17, 19 January 2013 (UTC)

I think part/all of this was my fault, although I don't remember the exact chronology. BLARGH. --Dfeuer (talk) 19:19, 19 January 2013 (UTC)

/* reverse, revert, go back, undo and desist */ This page was initially Transitive Closure Always Exists (Relation Theory) and expressed the fact that a transitive closure always exists. This has recently not only been renamed but also rewritten so that the redirect from Transitive Closure Always Exists (Relation Theory) points to a page which does not have that result on it.

The cardinal rule of this site (after the one that demands left/right braces on all parentheses) is that if a theorem is stated on a page, then you do not change the statement of the theorem. If you want a theorem that says something different, then write a new page. What may seem a silly thing to need to prove to you may be a crucial link in a complicated argument to another contributor.

With the above in mind, I am going to retrieve the original page that evolved into this one and restate it. This may take some time because the history of this page is somewhat convoluted. --prime mover (talk) 08:53, 7 March 2013 (UTC)

What instigated this is a reformulation of "Transitive Closure" as a closure operator with the recently added material on those general things. What may be appropriate is to rethink the definitions and consider whether all of them are actually sensible as a definition of the concept, rather than a mere characterization or construction.

It is certainly deeper than an intentional destruction of a page; what may be the underlying problem is lack of a mechanism to differentiate definitions from characterizations. Sources provide a good Ansatz to this but sometimes corners are cut and/or generalities aren't bothered with in texts geared to specific use. — Lord_Farin (talk) 09:08, 7 March 2013 (UTC)

Off topic: I had to google it to make sure of what it means - but it might be worth adding "ansatz" as a formally defined term on this site as it is more precise in the original German than any English translation, and it would be useful to be able to use it without needing to go off-site for help for readers. --prime mover (talk) 09:18, 7 March 2013 (UTC)

What one person understands one way, another understands another way. There's a certain amount of convenience to having equivalent definitions on one page. The original definitions of reflexive and symmetric closures were constructive, and the original definition of transitive closure was axiomatic. I filled in the gaps (or many of them, anyway).--Dfeuer (talk) 14:51, 7 March 2013 (UTC)

that's as maybe and the above paragraph possibly contains a grain or two of wisdom, but the point I'm making is that because of lack of care with the existing page structure, the page that was Transitive Closure Always Exists (Relation Theory) containing exactly that no longer exists. --prime mover (talk) 18:26, 7 March 2013 (UTC)

As there are constructive definitions, I would submit that it does not need to exist, any more than Square of Number Always Exists. --Dfeuer (talk) 18:38, 7 March 2013 (UTC)

But the innocent reader just coming in on $\mathsf{Pr} \infty \mathsf{fWiki}$ because their source, which uses an abstract definition, glosses over the matter would like that page to exist. Please keep this part of the audience (i.e. without full information) in mind. — Lord_Farin (talk) 22:15, 7 March 2013 (UTC)

I may have been influenced to large extent by the categorical perspective, where it is usually not the construction of something that is important, but the properties that it has (indeed, I think most of mathematics is about the latter rather than the former — again, I may have strayed far from the applied side of our beloved science). Introducing things "by construction" always has a certain kind of "educated guess"-feel to it; I generally like abstract things better. But this is not the place for such a debate. We shouldn't attempt to do too much at once.

It's just that abstract and constructive definitions don't go together very well; they necessitate results that are trivial from the abstract PoV, but not from the constructive and vice versa. For now, such is life. — Lord_Farin (talk) 15:01, 7 March 2013 (UTC)

Would you rather have them all defined abstractly, and then have pages transcluding multiple constructions and characterizations? I think the way it is now is easier to use. This would apply to, say, the real numbers too, where we would define them only axiomatically in your scheme (complete ordered field being one definition; there are a few more, I believe). --Dfeuer (talk) 15:08, 7 March 2013 (UTC)

I only stated some observations. I didn't intend to propose a paradigm shift. For the case of the real numbers, your suggestion is rather appealing to me. It then provides a convenient entry point for analysts not bothering with existence (i.e. willing to take it for granted), while the construction page would justify the definition based on various (likely mainly set-theoretical) more elementary schemes. From what I recall this is precisely how the concept of $\R$ is taught to freshmen. Admittedly there are some border cases (e.g. $\C$, which is usually defined as $\R \times \R$ with some specific operations rather than e.g. the abstract $\R[X]/(X^2+1)$) but usually it's quite clear whether something is viable as definition on its own or not. My general aim in these considerations is to avoid proofs "Follows a fortiori from $X$ being a letter" or whatever. As I said, I'd rather postpone this discussion to when our metaphorical paradigmatic sea has calmed down a bit. — Lord_Farin (talk) 15:40, 7 March 2013 (UTC)

Possible resolution: Perhaps on our definitions pages we should in some manner "tag" abstract definitions and constructive definitions and then in some as-yet-unclear-to-me fashion distinguish between proofs that certain constructions satisfy certain abstract definitions, proofs that certain abstract definitions lead to isomorphic structures, proofs that certain constructions construct the same objects as others, etc.... In some cases, there's only one object, as in transitive closure. In others (e.g., real numbers, metric or order completion, etc.) there are various isomorphic constructions and there is (somewhere, eventually) a proof that they are all isomorphic as ordered fields (or as topological fields, or as whatever else they may be of interest). --Dfeuer (talk) 22:29, 7 March 2013 (UTC)

Problem is, I have yet to locate the necessary creativity to come up with such a system, even though having contemplated it before. But your wording hits the nail on the head. — Lord_Farin (talk) 22:32, 7 March 2013 (UTC)

If it's unclear to you, then, as always, my recommendation is (no, not recommendation, my command) is: leave it alone. Such flashes of insight happen when they happen. It's no good to say: I'm uncomfortable with this approach so let's delete it and then er, try to dream up something, er, hum, it's not happening so, never mind, nobody will notice I deleted it, it was only written by some stupid fussy cross old man, anyway. --prime mover (talk) 22:35, 7 March 2013 (UTC)

Nothing wrong with stating the issue. I do agree with you that now is not the time to attempt to resolve this in whatever way thinkable. I already stated that above but I wanted to support your comment explicitly. — Lord_Farin (talk) 22:39, 7 March 2013 (UTC)

I wish to emphasize that NOTHING was deleted. The pages in question were renamed and modified SLIGHTLY to fit a different context. They are now near duplicates. --Dfeuer (talk) 22:49, 7 March 2013 (UTC)

What was deleted was the words "the transitive closure $\mathcal R^+$ of $\mathcal R$ always exists." That is deletion, is it not? The most important words on the entire page? --prime mover (talk) 23:11, 7 March 2013 (UTC)

Dig your trenches please. And make them so deep you can see nor hear the other side. TIA. — Lord_Farin (talk) 23:12, 7 March 2013 (UTC)