Definition talk:Real Interval

I've been wondering what others think about the double dots notation. Is this common enough in texts these days that the average person who is interested enough in math to be using proofwiki would recognize it? --Cynic-(talk) 21:48, 26 December 2008 (UTC)

I'd say stick with using a comma. --Joe 21:56, 26 December 2008 (UTC)

I'd say whatever you use, make it obvious what it means. It's easy enough to get confused when reading $(x, y)$ to think it may mean a coordinate pair, but (once explained) the two-dots notation is unambiguous.

When you're writing a proof, it's usually obvious to a writer what all the symbols mean, but I don't think it's reasonable to expect anyone coming upon a page from cold should have any "prior experience" of anything in the proof at all.

As for two-dots v. comma, if we are going to start using all the most up-to-date notation, then we're going to use two dots. If we're not going to be consistently using the most up-to-date notation, we should be able to continue to use, for example, $I_S$ for the identity mapping on the set $S$ rather than the horrible $1_S$, and carry on using $e$ for the identity of a monoid rather than the equally horrible $0$ or $1$, which are currently fashionable, apparently. --Matt Westwood 23:24, 26 December 2008 (UTC)

I noticed something when I was going through this: Suppose $a, b \in \R$ comes before the ToC, which isn't good. Any thoughts on how to fix this? --Cynic-(talk) 02:34, 27 December 2008 (UTC)

How's that? --Matt Westwood 11:33, 27 December 2008 (UTC)

Much better, thanks! --Cynic-(talk) 17:53, 27 December 2008 (UTC)

On the definition of real intervals
Isn't it much more elegant to define intervals as in Interval Defined by Betweenness, rather than define it as one of 10 (!) types? This avoids as well to use the vague word "between" in the sentence
 * The set of all real numbers between any two given real numbers $a$ and $b$ is called a (real) interval.

From the informal definition, it is not clear whether "between" includes endpoints. Aren't informal definitions meant to make things more understandable? :)

I could write out a proposal here on the talk page for an alternative organization. What do you think? --barto (talk) 10:19, 28 January 2017 (EST)


 * Well of course you don't know from the informal definition what sort of interval you are talking about, because "betweenness" applies to all intervals. Which is all very well, but most cases you do need to know what kind of interval you are talking about. Hence the 10 different types of interval. --prime mover (talk) 12:00, 28 January 2017 (EST)


 * I agree we should define each of the types, but what I said is I wouldn't take them as definition of "interval". I.e. rather than "A real interval is any subset of $\R$ of a type listed below." I propose to define it like: "An interval is a subset $S$ of the real numbers with the property that $x,y\in S$, $z\in\R$ and $x<z<y\implies z\in S$." And then say: "In Interval Defined by Betweenness, it is shown that every interval is of one of the types described below." --barto (talk) 12:10, 28 January 2017 (EST)


 * This way, we have a concise but formal definition, rather than a definition that's 200 lines long :) --barto (talk) 12:13, 28 January 2017 (EST)


 * Yes but it's not always $x < z < y \implies z \in S$ (spaces in your $\LaTeX$ are good btw) because sometimes your $<$ should be $\le$, and that makes it a different interval. --prime mover (talk) 15:13, 28 January 2017 (EST)


 * Ehm I think I did not make myself clear. I don't want to define an interval as a set of numbers $z$ satisfying $x<z<y$ for some fixed $x,y\in\R$ (which would indeed be incorrect). I want to define it as a set $S$ satisfying: $\forall x,z \in S : \forall z \in \R : (x < z < y \implies z \in S)$. The same as in Interval Defined by Betweenness. --barto (talk) 15:19, 28 January 2017 (EST)


 * I am not sure if definition should be moved to a completely different page, but maybe this one could be improved by not explicitly stating all possible cases, or at least not of equivalent importance. Say, every interval can be empty/nonempty, then every nonempty can be bounded/unbounded, and so on. This way the number of definitions on this page is reduced. Like we do not write down definitions of all real functions according to a type of domain they exist in and a differentiability class they belong to on the same page, in the same way here some definitions could be contracted into categories denoting some additional constraints on the broader description. But if it can be shown that Interval Defined by Betweenness is enough to list all listed intervals as special cases, then the definition should be moved. Julius (talk) 16:18, 28 January 2017 (EST)


 * I just can't see where the improvement would lie. As it is now, it is clear what each interval is. Using branches and sub-branches and sub-sub-branches brings to mind some of the appallingly bad software that I am losing my health over in my hated and despised day job. --prime mover (talk) 16:28, 28 January 2017 (EST)


 * No, I'd certainly not remove any definitions. Let me make my point clear: on that page, the concept real interval is defined. It does so by defining it as one out of a list of sets, which makes for a very lengthy definition. My point is that a two-line definition is much easier to read. People might be curious to see how intervals are defined at ProofWiki (e.g. because they want to dive into the foundations of analysis - I see no other reason why anyone would visit that page), but they won't be happy if they have to read a lot to get their answer. Short definitions are much more user-friendly.


 * Removing definitions of interval types or describing them would make the definition of real interval even more difficult to look up (not to say make invalid). --barto (talk) 16:32, 28 January 2017 (EST)

"Short definitions are much more user-friendly." I strongly disagree. Rigorous definitions are user-friendly. Oh, the countless times I have lost track because someone thought "let's cut out the explanation and just give the formalism"... E.g. Principia Mathematica.

However, it is true that this page currently does not, strictly speaking, include a paragraph defining "interval" as such. It only describes types of intervals. I think it would add value to include the definition based on betweenness. Prerequisite to this, as always, but particularly with such a fundamental definition, is sourcing it as such in the literature. &mdash; Lord_Farin (talk) 04:46, 29 January 2017 (EST)


 * I've learnt by now that the correct approach is indeed to give both definitions :) I have tried hard to find the other def in literature, haven't found it, only in course notes. --barto (talk) (contribs) 05:35, 17 November 2017 (EST)