Definition talk:Ordered Set

Don't know if this would be instructive: A mention that this specifically does not include a preordered set.


 * No, because an ordered set is a priori a preordered set.


 * We can link to preordered set if you like, but your suggestion is like saying "a mention on Definition:Rational Number that this does not include an Definition:Integer. --prime mover (talk) 12:36, 7 March 2013 (UTC)

Classes again
The nice thing about "ordered structure" is that it doesn't have the word "set" in it. What do we call a relational structure whose relation is an ordering on a class? --Dfeuer (talk) 22:48, 21 March 2013 (UTC)


 * The disadvantage to "ordered structure" is that it usually means something other than "ordered set". "What do we call ... ordering on a class?" A stinking bodge that deserves to be flushed down the toilet.


 * But what we do is the hard work underpinning the basics of what L_F was talking about before we embark on the fun stuff. If you're really interested in working towards a solution to this question that you have jumped in the middle of, then rather than worrying about trivial issues like names of stuff, you'd be establishing how to set up a technique whereby multiple axiom schemata can be paralleled. --prime mover (talk) 23:10, 21 March 2013 (UTC)


 * For now I will be working on this stuff in my sandbox. I don't know how to deal with parallel systems. A rigid approach would be to put each axiomatization in its own namespace and only allow theorems about *sets* (not classes) in (Main), but that seems a bit draconian, especially considering that order theory and (I'm given to understand) category theory often deal in proper classes. Unrelatedly, I want to re-raise my request for a Physics (or perhaps Science) namespace. --Dfeuer (talk) 23:23, 21 March 2013 (UTC)


 * And if you think everything to do with class theory deserves to be flushed, I'm afraid for your sake that there seems to be very little chance of that. Classes have been a subject of mathematical study for decades. Of course, if you can prove that NBG (or even Morse-Kelley) theory is inconsistent, you could possibly be on your way to getting your wish (and the respect of mathematicians around the world). --Dfeuer (talk) 23:27, 21 March 2013 (UTC)

There's currently no analogous definition for classes, this proof references it and there isn't a page to link it to. I'd also like to extend this to be for classes so that well orders can be induced on proper classes by bijecting them with the class of ordinals via limitation of size, but I currently can't as there's no corresponding definition. --TheoLaLeo (talk) 01:40, 24 November 2021 (UTC)


 * I see you're taking on the Jech work. I presume it has an "ordered class" approach. So if you want to take on this exercise, go ahead. My advice is to attack this using Jech as your source work, so as to make sure it's apporopriately rigorous and sourced.


 * Incidentally, the approach to linking where you use this and this is suboptimal because the reader can't see what you're referring to unless they either click on the link or edit-source the page. It's practically always far better to show the entire link, in this case: Strictly Increasing Mapping on Well-Ordered Class and Injection Induces Well-Ordering. I can see no advantage to hiding the title behind a "this". --prime mover (talk) 08:33, 24 November 2021 (UTC)


 * Jech surprisingly takes the usual ordered set approach, but skimming through Book:Raymond M. Smullyan/Set Theory and the Continuum Problem, he seems to give an approach that works for classes on page 43. I may try to work from him. --TheoLaLeo (talk) 08:53, 24 November 2021 (UTC)


 * See what I do with books I'm working through. I copy the Sources link from the last page I got to in any particular work into the page of that book (note we are using the $2010$ revised edition Book:Raymond M. Smullyan/Set Theory and the Continuum Problem/Revised Edition, that's the only one on my bookshelf) so we know how far we've got. Careful about the onlyinclude tags, they are so I can maintain a general page of source work progress of the works on my shelves. I got to about page $25$ in April 2020 but ran out of steam and needed a break, and worked on something else in the meantime. Kept meaning to get back to it but I need to have no other distractions or I lose the thread.


 * If you have the revised $2010$ edition, then feel free to follow it through and see whether you are in a position to pick it up and run with it. If it's the $1996$ edition you've got, then feel free to process your way through that (we have a precedent for working through multiple editions of source works, it's particularly interesting when consecutive editions vary markedly) and we can work through these things in parallel. --prime mover (talk) 09:14, 24 November 2021 (UTC)