Definition talk:Strong Well-Ordering

Are we going down the route that suggests that every statement made in set theory has a "strong" version that applies in class theory? --prime mover (talk) 21:22, 25 April 2013 (UTC)

More to the point, where is your citation for the suggestion that such statements are known as "strong" versions? Sounds fascist to me. --prime mover (talk) 21:23, 25 April 2013 (UTC)


 * No citation for that. I have only sources defining well-ordering as one and sources defining well-ordering as the other. I will expand the note. We probably don't need "strong" forms of very many things, actually. Many/most concepts translate without any complications. Well-foundedness and choice do not behave so well in that regard. --Dfeuer (talk) 21:54, 25 April 2013 (UTC)


 * This is an example of what has been discussed many times before. We are not in a position to tackle class theory. We are not going to do it yet. By "we" I specifically and incontrovertibly mean YOU.


 * Until we have fully established the fundamentals of class theory (based I suppose on BGN) then we can not come out with bald definitions of class-theoretical versions of statements of set theory as "strong" versions of those concepts. --prime mover (talk) 05:18, 26 April 2013 (UTC)


 * We already have plenty. "Class theory" as you call it is a major approach to set theory. I understand you believe I am some sort of moron, but that is not the case. --Dfeuer (talk) 08:25, 26 April 2013 (UTC)


 * The merits or not of class theory are not the issue here. The issue is the terminology. We have no source for "strong" in the terminology, it's just a word (descriptive or not, again that's not the issue) you've pulled out of the air. Your intellectual capabilities, again, are not an issue here; what *is* at issue is the fact that we are not in a position to start dictating terminology, *particularly* with such a "strong" word (sorry) as "strong".


 * What *is* at issue is your lack of ability to see that this is a concern, and your insistence of taking this as a personal attack on your intellectual integrity. The consequence is that you go ahead and defy the instructions given because in your selfish childish arrogance you know better. THAT is what is at issue here. --prime mover (talk) 11:21, 26 April 2013 (UTC)


 * Again, the problem is that sources coming from different perspectives use the same terms to mean different things. So we have to deal one way or another. We could name the page Definition:Well-Ordering/Subclass Minimum or some such, I suppose, if that would match your peculiar needs better. I think we get more descriptive and less deceptive theorem titles if we use a different name, though. --Dfeuer (talk) 19:50, 26 April 2013 (UTC)


 * No, what we'll probably end up doing is going through and deleting the lot, then starting from scratch and doing it properly. --prime mover (talk) 22:08, 26 April 2013 (UTC)


 * It's a lot easier to edit and move than to delete. There's nothing wrong with it except your distaste for the name. --Dfeuer (talk) 22:15, 26 April 2013 (UTC)


 * There is more to it than that.


 * The whole approach to class theory needs to be redesigned from ground up before we can start thinking about how to craft the stuff which you're making a valiant but misguided attempt on. This has been explained time and time again: please do not work on this area until it has been so worked over. --prime mover (talk) 22:27, 26 April 2013 (UTC)


 * "It's a lot easier to edit and move" -- all right for you to say, you're not the poor muggins who ends up cleaning up your rectal incontinence. You have been informed: unless you are going right back to first principles of class theory and working through from scratch (and by first principles I don't mean Smullyan and Fitting, I mean Bernays) then you will be blocked from this site for a period of no less than one year, by which time you may have learned to respect authority. --prime mover (talk) 22:27, 26 April 2013 (UTC)

Huh? We haven't put up Newton's, Leibniz's, Russell's, Peano's, Zermelo's, or Fraenkel's writings yet... should we get rid of everything to do with calculus, set theory, and arithmetic site-wide? That's no way to get anywhere. I understand you don't like me, but that's no reason to be unreasonable. --Dfeuer (talk) 23:11, 26 April 2013 (UTC)


 * You are either not reading what is being written, failing to understand what is being written, or being disingenuous.


 * This specifically applies to class theory. The fact is that the boundaries between class theory and set theory are blurred. Some theorems/definitions are specifically applicable to classes only, and some to sets only. Some are applicable to both.


 * That is correct.


 * Now it has not been established which are which. On this site, at least, there has been no attempt to do this - all that has been done is that some subset of the pages of the as-far-as-it-goes set theory work have class-theory versions of them (in that they are exactly the same but with "class" in them and pompous self-important titles which include the word "strong", for example). Others have just had "or class" inserted into the pages at appropriate places. Both approaches are severely suboptimal.


 * The two approaches address completely different situations. There is a very large category of theorems whose "obvious" proofs are identical whether the collection being considered is a set or not. Every theorem using only empty set, binary union, binary intersection, set/class difference and subset/subclass will have precisely the same proofs. A large portion of the basic theorems on infinite unions and intersections will go through unchanged, although they are no longer strictly more general. Most texts, papers, etc., will not bother to go into full detail about each of these simple theorems. Smullyan and Fitting do recommend a particular source (I'll check which) that covers NBG in a more formal way—perhaps that would be more to your liking, but I don't know if even they will go into quite that much detail on such basics. The other sorts of things (where I angered you by using the word "strong") are things that look very similar but are different in some significant way. The difference between what I called "well-founded" and what I called "strongly well-founded" is very much analogous to the differences between "lattice", "$\omega$-complete lattice", and "complete lattice". In the realm of countable lattices, $\omega$-complete is the same as complete. In the realm of general lattices they are not the same. Similarly, a relation on a set is strongly well-founded iff it is well-founded, but that does not necessarily hold for relations on classes.


 * Either the works from which the class theory pages have been sourced so far are deficient in this regard, or the contributors using those sources are insufficiently skilled or motivated to perform an adequate task of extracting the information and presenting it in a consistent manner compatible with the format that has evolved on this site. Either way we do not want to continue the way we are doing.


 * Before going any further down this route it is essential that a viable approach is determined. Because of your proven track record of inability to cooperate with convention, your contempt for house rules and your lack of patience with finishing off even the most basic of contributions, you, personally, are not the person who is going to do this.


 * I am, despite all this fighting, still willing to work cooperatively to figure such things out.


 * Unreasonable or not, this is the way it is. Any more work from you on class theory will mean you are going to get blocked. --prime mover (talk) 07:42, 27 April 2013 (UTC)


 * We are here to do math. --Dfeuer (talk) 15:55, 27 April 2013 (UTC)


 * Actually, we are here to develop a well-structured and rigorously formatted dictionary of mathematical proofs. Any fool can "do math", if that's all you want to do then goodbye. --prime mover (talk) 20:53, 27 April 2013 (UTC)


 * So we're agreed then: you lay off class theory for the moment, on pain of being blocked for a year? Good. --prime mover (talk) 17:09, 27 April 2013 (UTC)


 * I'm going to try to ignore your bullying and see if it stops. On a practical note: I took a look at the beginning of Bernays' development, and I think you severely underestimate the amount of pain that will be involved in doing it exactly as he did in the thirties, while keeping everything rigorous:


 * The simple statement $x \in A \subseteq B \subseteq C$ in a common style (S&amp;F might or might not capitalize the $x$, depending on mood, and Kelley would put it all in lower case) becomes, in Bernays, eight different possible statements:


 * $x \mathrel\epsilon a \subseteq b \subseteq c$
 * $x \mathrel\eta A \subseteq b \subseteq c$
 * $x \mathrel\epsilon a \subseteq B \subseteq c$
 * $x \mathrel\eta A \subseteq B \subseteq c$
 * etc.
 * I imagine that it is probably somewhat easier to formalize it the way he did, but there is a reason that a number of later writers dropped it. In the context of, that kind of explosion would be nightmarish. --Dfeuer (talk) 23:25, 27 April 2013 (UTC)


 * Yes I know it's going to be difficult. This is why I don't want you doing it, because you're not capable. --prime mover (talk) 10:34, 28 April 2013 (UTC)


 * You're either not reading what I've written, failing to understand it, or being disingenuous. My ability has nothing whatsoever to do with it. Bernays' original, ground-breaking formalization is simply not suited to, while later refinements by others are. Bernays wasn't writing a textbook on set theory; he was building a finite axiomatization of set theory. Ease of use and consistency of representation simply were not relevant to his purpose. --Dfeuer (talk) 16:08, 28 April 2013 (UTC)


 * "Bernays' original, ground-breaking formalization is simply not suited to " ... You of course now this from your many years experience with both class theory and ProofWiki. Yeah right. --prime mover (talk) 18:53, 28 April 2013 (UTC)

Historical note: 1940. Gödel and Brown. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis already drops the wall, using consistent notation for sets and classes and giving an axiom that every set is a class. --Dfeuer (talk) 18:53, 30 April 2013 (UTC)


 * In fact Gödel-Brown looks like it could be a very good approach for our purposes. It begins at the beginning much more than Bernays does, and seems over all quite clear, careful, and readable. If you're willing to accept a text by a founder of class-set theory other than Bernays, I think this could be a good choice. It is a set of notes by Brown on lectures by Gödel, corrected and revised several times by Gödel. --Dfeuer (talk) 19:47, 30 April 2013 (UTC)


 * That's as maybe, but I don't want you doing it, you lack the level of discipline required. --prime mover (talk) 20:03, 30 April 2013 (UTC)


 * Nobody's perfect. If I do it I'll try to build it in userspace until there's enough to be worth something. It would be awfully nice to get to the point where we can put up a proof that NBG is a conservative extension of ZF. --Dfeuer (talk) 20:28, 30 April 2013 (UTC)