Talk:Union Distributes over Union

Classes or sets: why?
There's the thought of merging this with the "equivalent result" for sets. Now we've had this conversation before, over and over again, and I have not got to the bottom of it: at what point does it become important that a result holds for classes as well as for sets? What advantages do we get by stating exactly the same result in set theory as in class theory? Who is prepared to go through every single result in set theory and copy it to a parallel result for classes? Finally, what results do we specifically need to prove in order to be able to do things in class theory that you can't do in set theory? --prime mover 21:29, 4 August 2012 (UTC)


 * Classes overcome the trouble posed by Russell's Paradox. However, they themselves are a very valuable source of structures, and I think most of the stuff in the abstract algebra section can immediately be generalised to taking an underlying class instead of a set. Other than that, the issue is merely foundational. Ultimately every result holding for classes holds for sets, sinds the latter are classes themselves. The converse does not hold, 'power class' is a void expression (in the simple Gödel-Bernays axiom system). This can be superseded again by inventing $n$-classes for each integer $n$, for each ordinal $n$ and so on and so forth. Before I lose myself too much into the foundational aspects of the approach, I think that ultimately the reason for stuff like this cropping up is that ProofWiki disallows saying 'Trivially, theorem Bla extends to classes' or 'Mutatis mutandis, this result applies to classes as well'. In general the solution would be to prove results only for the more general concept, and then possibly refer to it (you have done this yourself an infinitude of times in the abstract algebra section), but for the uninitiated word set is much more comforting than the word class.
 * In summary, the results for classes (and in the future, more general things encompassing classes that also support power-'general things') force us to face the fact that the strict rigour strived after on PW leads to apparent humongous multiplication of results. One of the things I have tacitly found myself applying the last few days is to loosen the very strict rigour policies so revered by all of us, in order to allow some intuition to come to play when discussing some more intricate proofs. This is the only feasible way to go if we can't come up with principles as general as e.g. Principle of Duality of Huntington Algebras to accommodate for the generalisations' justification. There is however a problem with the applicability of such principles as they generally require that one know precisely which of the set theory axioms are employed in a particular proof; this is unwieldy if we can't let a computer determine it for us, and even then it won't be easy.
 * With all this negativity spit out, let it be remarked that a lot of the current research in logic and foundations focuses on developing ways to determine which proofs depend on which axioms, generally by developing new systems with fewer or more axioms and investigating their truth sentences in some appropriate deduction system.
 * Ultimately, classes are not but a similar venture in that direction, and the problem we face is to determine and clearly indicate the subset of theorems in set theory that also hold for classes. The staggering dimensions of such contemplations place us in philosophical discussions on the direction of the site and ideas about automated proof systems, and ensure that lengthy debates such as this one are inevitable because there simply isn't a conclusive and concise answer (as of yet). I could write a lot more but let me try to refrain from frying our brains. --Lord_Farin 22:02, 4 August 2012 (UTC)


 * You have voiced my own concerns. Yes, I understand that "they (classes) themselves are a very valuable source of structures", but beyond seeing abstract statements saying that and similar things, I have actually seen little practical application of this. And besides: with ZFC you can define everything you need in order to get anywhere you want in "conventional mathematics": the progression from there via Peano to reals is documented fairly thoroughly here. But until we specifically state exactly what you can and can't port across to class theory from set theory ... no I don't know what I'm talking about - it's late and I have an early start in the morning and about 4 projects to finish tomorrow before I can even think about ProofWiki. I will just have to shelve this and let the class theorists get on with whatever they're doing and be damned to the structural integrity of the site. --prime mover 22:23, 4 August 2012 (UTC)
 * I admire your determination to keep on slogging in the tidy section; I have tried it for some time but I always get the urge to contribute more substantial things. Fortunately (well, that's rather relative) the base of active contributors is very small so that it is not necessary for me to get completely bogged down in tidying others' additions to the site. OTOH I have a tendency (as you undoubtedly noticed) to simply put a tidy tag and leave as fast as I came. Cursed laziness and short attention span. 'gnite. --Lord_Farin 22:48, 4 August 2012 (UTC)
 * Cursed short attention span and laziness on the part of certain contributors not being bothered to (a) learn the house style nor (b) try and fit new stuff into the context and framework of what's already there makes me come very close to just deleting the entire bags of shit without even bothering to read them. As you see I'm starting to just bung a tidy tag in myself. These new pages on ordinal arithmetic will one day get fitted into the existing pages on ordinal arithmetic, but for now I'm just going to pretend they don't exist. --prime mover 05:28, 5 August 2012 (UTC)
 * There are existing pages on ordinal arithmetic??? I have not found them this whole time!  I have searched, too.  Also, how do you make indexed unions look "nice" in the eqn format?Andrew Salmon 14:21, 5 August 2012 (UTC)


 * Fact is, there's a lot of overlap among results in the "Ordinals" and "Ordinal Arithmetic" categories, and also many of the results in the various structures defined in the various "Natural Numbers" categories. The latter contains an attempt to demonstrate that the ZFC axioms (specifically the Axiom of Infinity) allow the definition of a structure which fulfils the Peano axioms, which then allow the definition of the natural numbers. (There are also various other axiomatic derivations of the natural numbers.) The work done in Ordinal Arithmetic is very much paralleling this latter exercise - except that with the introduction of the language of "classes", the one can not be directly interspersed with the other as ZFC is specifically a set based axiom system. The worst approach to this would be to create two completely independent sets of results: one for sets, one for classes, both completely independent and with their own idiosyncratic symbology. --prime mover 16:07, 5 August 2012 (UTC)


 * Ordinal Arithmetic can sort of be regarded as a generalization of Natural Number arithmetic, but it is very distinct. For example, it is non-commutative -- $\omega + 2 \ne 2 + \omega$.  I believe that the results should be kept separate.  As for Ordinals, there are some overlapping results, especially between strict well-ordering and wosets, but other than that, most of it is (from what I have been able to find on the site) new material.  I understand your frustration about sets/classes, and I'll try to use sets when possible, since that would be more consistent with the material already on this site.  This page should be moved to Union Distributes over Union, to be consistent with the rest of the site. --Andrew Salmon 16:20, 5 August 2012 (UTC)


 * I think we still need a set of results which state that for finite numbers, ordinals and cardinals behave in the same way and addition and multiplication are commutative. Then at least we will be able to just rely on one set of results for that limited domain. Hell, the maths itself is so unnecessarily messy in this area to the extent of being hateful. --prime mover 16:40, 5 August 2012 (UTC)


 * The maths is messy since it is in fact a lot of fields coming together on a common intersection. The same stuff can be phrased in an infinitude of ways; since we want to document them all, it follows that we have to accommodate for (part of) this infinitude, and hence have to face the technical difficulties coming along. There is probably a reason for authors confining themselves to only one system -- nonetheless we still need to pursue our full documentation aspirations. --Lord_Farin 16:47, 5 August 2012 (UTC)

Just yesterday I amended the Help:Editing/House Style page to make other contributors aware of this. To achieve the spacing around $\in$, one simply inputs e.g.  as opposed to   inside the subscript. At least, my guess is that that was your question. --Lord_Farin 16:12, 5 August 2012 (UTC)


 * Thank you. This partially answers my question.  But see Limit Ordinals Preserved Under Ordinal Addition.  Notice how in the line:


 * The $z$ is a little bit higher than the $\implies$ sign, and the whole line seems to be shifted slightly upward. Is there a good way to deal with this?  Is this a problem at all? --Andrew Salmon 16:28, 5 August 2012 (UTC)


 * This is a consequence of the material in either cell (the $\implies$ and the rest) being centered vertically. This problem is currently not solvable. I may think of implementing a fix to this aesthetic issue, but such will be HTML/CSS related and so will be implemented (if ever) for all pages at once. From your side, no action is required at this point. --Lord_Farin 16:31, 5 August 2012 (UTC)