Talk:Set Equivalence behaves like Equivalence Relation

This theorem is too broad as stated to match Definition:Equivalence Relation
Definition:Equivalence Relation defines equivalence relations on sets. The statement of this theorem makes no mention of the set involved. So either this needs to refer to a broader definition of equivalence relation (allowing relations, e.g., on the (proper) class of all sets), or the theorem needs to be narrowed to say that if $S$ is a set of sets, then set equivalence is an equivalence relation on $S$. --Dfeuer (talk) 04:01, 12 December 2012 (UTC)


 * I vote for both, but while I like the first one better, it must be remarked that under current terminology it isn't strictly speaking an equivalence. --Lord_Farin (talk) 15:48, 12 December 2012 (UTC)


 * I vote for the second for the reason L_F mentioned. Please be very clear on this page if you have to change it and use the concept of equivalence class. Although it should be done in this case I question the usefulness of setting up a UoD for everything. Please do not get carried away and change theorems like Union is Associative to include statements like:


 * $\forall A,B,C \in U: A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$


 * --Jshflynn (talk) 16:58, 12 December 2012 (UTC)


 * Sadly, I don't understand either answer to my question. Lord Farin, why is it not an equivalence? Jshflynn, the definition of equivalence class on this site (currently) is defined in terms of equivalence relations, so that doesn't give any extra strength. --Dfeuer (talk) 18:09, 12 December 2012 (UTC)


 * An equivalence relation is defined as a binary relation on a set $S$. One suggestion you made was that we introduce set equivalence as a binary relation on the proper class of all sets. So technically it wouldn't be an equivalence relation. I believe this was LF's point.


 * My point was to be careful with the term "equivalence class" if you had to use it in your reworking of the article. This is because the it contains the word "class" and so could cause confusion.


 * After some thought though I'm inclined to agree with Prime.mover about the whole thing. --Jshflynn (talk) 18:40, 12 December 2012 (UTC)


 * I don't think the discussion is profitable in the first place. It just adds unnecessary complication to a simple concept which is easy to grasp. --prime mover (talk) 18:22, 12 December 2012 (UTC)


 * It could be extrapolated from your comment that you deem axiomatic set theory "unnecessary complication" from naive set theory. We are dealing with something that looks like an equivalence relation on something that is not a set. There are two options: amending this page, or amending "equivalence relation" to be defined on classes rather than on sets. How about expanding PW into fields of mathematical research as of yet uncharted first, and returning to this minor issue in a few aeons? --Lord_Farin (talk) 18:58, 12 December 2012 (UTC)


 * I have philosophical issues with the whole concept of class theory. It's messy. It's like putting a band-aid on gangrene. It's inelegant so therefore has to be wrong. Unfortunately I have no alternative approach.


 * In the meantime, in order to ensure that our set theory statements remain accurate and of maximum applicability, it seems that they all effectively need to be amended to apply to classes as well as sets. But because it is only in rare places that the distinction between sets and classes needs to be made, the whole class superstructure feels like (to use a business analogy) a top-heavy layer of project management.


 * But if you really feel strongly about adding class theoretical extensions to set theory stuff, then you'll probably go ahead with it rather than just talking about it. --prime mover (talk) 20:05, 12 December 2012 (UTC)


 * Well, there are alternative formulations of set theory that don't use them, but those seem to be considerably less popular. My conjecture is that very few theorems that apply to proper classes are actually used in that context, but that this theorem is one of them. --Dfeuer (talk) 21:27, 12 December 2012 (UTC)