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Smullyan and Fitting
Note the pages that I have added containing the S&F work: the link you want to add at the bottom of your pages is now. Have fun.

How to treat the issue of class vs. set in all the pages which have practically identical contents for both set and class has been vexing us for a long time.

1) It would be good to have "purely set theory" pages wherein the issue of classes does not appear at all, so as not to cause confusion for those who do not know what a "class" is or why it is important (or indeed, for their purposes, whether).

2) On the other hand we do not want to duplicate the entirety of the set theory category with just "set" replaced by "class".

Your compromise is as good as any other we've devised but it still does not feel optimal. The whole concept of "class theory" sucks cat farts out of a lead balloon anyway - the whole shebang is a messy kludge to get around Russell's paradox and I'm afraid I cannot be party to it and keep what little remains of my sanity. I have the Bernays work - one day I'll transfer the contents to ProofWiki but that won't be anywhere near immediate. --prime mover (talk) 21:48, 21 March 2013 (UTC)


 * Avoiding self-contradictory axioms seems to be rather important. I'm not bothered by putting sets and classes side by side. What's more troubling, from my perspective, is that different theories have different notions of the relationship between sets and classes. Smullyan and Fitting decree that every set is also a class. Some others apparently do not, preferring instead to have a notion of "small class", a class that is extensionally equal to some set. This leads to a bit of a clash in terminology. Another terminological issue: the current definition of a foundational relation is a weak one, requiring only that every non-empty subset have a minimal (or initial) element. Smullyan and Fitting instead require (of a "well-founded relational system") that every subclass have an initial element. These are apparently equivalent if the axiom of foundation is accepted. The intuitionists apparently prefer a still stronger definition, requiring that it be possible to do induction relative to the relation. This is classically equivalent to Smullyan and Fitting's definition, but apparently not intuitionistically. Perhaps the stronger one should be "strongly foundational/strongly well-founded"? --Dfeuer (talk) 22:07, 21 March 2013 (UTC)


 * Not my business but I will just say it anyway. Why not just implement a principle? Have a look at: Duality Principle (Order Theory) for example.


 * The principle you make will probably be more intricate than that but it will save you a lot of, frankly, artificial work.


 * Mathematics really is like a tree sometimes. Its roots branch out too. --Jshflynn (talk) 22:28, 21 March 2013 (UTC)


 * TL;DR: isn't ready for this. More contemplation with broad (indeed, site-wide) perspective is necessary.
 * So far we have been unable to come up with a section where multiple genuinely different (i.e. not equivalent through some perhaps difficult theorem) approaches are taken. The PropLog section is the most advanced in this regard, and were it not for my reprioritisation I would be working hard on accomplishing this feat (described by some, upon success, as "something seriously worthwhile"). Because the logic section is more suited to this kind of exploration (logicians deal with different logical systems all the time; set theorists or mathematicians building on set theory usually bother with precisely one version of set theory) I suggest that further explorations in this direction are postponed until we have generated a proper framework (e.g. support for proving that "theories are equivalent" when they are formulated in different signatures) which will necessarily see its first fruits in said logic department. Just my 2 cents. &mdash; Lord_Farin (talk) 22:32, 21 March 2013 (UTC)


 * @Jshflynn: Problem is that the (more precisely, some) different formulations of set theory are not equivalent. This prohibits as simple a scheme as you set out. &mdash; Lord_Farin (talk) 22:32, 21 March 2013 (UTC)


 * The text I'm currently going through deals with NBG class-set theory (a conservative extension of ZF(C)) under classical logic and classically-embeddable modal logic, both with and without foundation, replacement, choice, CH, and GCH. So there's no major conflict between this and classical ZFC. However, there are (1) generally unimportant but annoying differences between theories such as whether a set is a class and (2) classically equivalent but non-foundationally, intuitionistically, or constructivistically distinct definitions of certain terms. We need to figure out how to handle such sort-of-equivalent-but-not-really definitions. --Dfeuer (talk) 22:44, 21 March 2013 (UTC)


 * Exactly L_F's point. Seriously, while it's a passable stopgap to add "... (or class)" to wherever a set is mentioned, it is at best a compromise and a not-very-good one at that. I'm having similar trouble getting the whole area of polynomial theory structured in some way so that the theorems and definitions all mean what is intended in each of the various contexts in which they are raised - and nothing hits the spot.


 * You understand better now the reasons behind my insistence on sourcing all (at least) definitions and axioms from hard-copy sources? --prime mover (talk) 23:02, 21 March 2013 (UTC)


 * First, LF is correct. In the long run it would be better if the machinery for comparing theories by strength was developed first.


 * In reality though has to grab what it can get from volunteers and Dfeuer is interested in this it seems. @Dfeuer You seem to have a large perspective on this. Can you list all the distinct theories (and their major source works if possible) that 95% of contemporary pure set theorists actually work with please.


 * No, I cannot. I poked around to try to get a sense of things, and the main sense I got was of a whole zoo of different theories and philosophies and so on. The source-rigidity policy actually does not help here at all—in fact it's more of a hindrance. When three different sources define the same term to mean three different things, there is value to being able to give them three different names. See, for a simple, example, the debate on Locally Compact—there's a traditional definition, and then there's a stronger sense that's equivalent for Hausdorff spaces (Munkres calls that "strongly locally compact" but acknowledges no standard term for it). On that page, PW chose the stronger sense and only mentions the weak sense in an "alternative definition" on the same page. Not ideal, really, but the conversation fizzled with no resolution. Aaaaanyway. The real point I think is that sources are a zoo and just being sure to stick with them doesn't give any guarantee of coherence. --Dfeuer (talk) 23:16, 21 March 2013 (UTC)


 * Concur on the zoo part. However I do feel that using sources is more likely to provide coherence than obscuring the origin of a train of thought would (in that source links provide a means to let multiple pairs of eyes assess intention, validity, etc. of a particular source). In the utopian limit, sources could all be sifted through and a true amalgamate of sources and mathematical knowledge would replace . Sadly, as-is this will require an infinitude of hard work. Nothing wrong with hard work. In fact, I thought I acknowledged the difficulty of the present process; in any case, glad you agree on that one. But difficulty alone is not a conclusive reason to abandon the project. That's it for now &mdash; LF out, off to bed and thesis (in that order). &mdash; Lord_Farin (talk) 23:33, 21 March 2013 (UTC)


 * Not all sources are created or looked upon equally. A wiki has absolutely no reputation at all. It is dependent on its sources. You already know this of course but I just feel I need to say it out loud for some reason. --Jshflynn (talk) 23:30, 21 March 2013 (UTC)


 * In that spirit I'd like to add that anything dependent on its sources needs to be judicial in choosing them. There's a real tendency to look on books and articles as reliable sources, simply because of their format. I think it's important to remember just how easy it is to get any kind of work published somewhere. --Linus44 (talk) 17:37, 22 March 2013 (UTC)

It's also important to consider the ages of sources. As mathematics develops, mathematicians develop new views of what constitutes the essence of an idea. A reliable, or even once-definitive, text may not be the best source for definitions if it is not very recent. Similarly, an introductory textbook (or even, perhaps, some advanced ones) may not be the best source for definitions because there may be subtle limitations that don't surface in that context. --Dfeuer (talk) 18:08, 22 March 2013 (UTC)


 * No, I'm serious. I understand that some contributors look upon ProofWiki as a learning experience, as in: "if I post up loads of maths then maybe some of it will sink in" but that's not what this was originally about. If you are really having difficulty getting your head round an area of mathematics, then you are encouraged to leave it alone as you are by definition not necessarily going to do a good job on it.


 * I think you need to read what I write more carefully. The problem we are discussing here has nothing to do with my difficulty, or lack thereof in any field of mathematics. It has to do with problems inherent to the task of putting together many different sources with wildly divergent perspectives based on different understandings of similar topics. As for the rest, I suspect that if you limit contributors on PW to experts that you will find there are very few, if any, people remaining.


 * You were the one who complained that the exercise described above was too difficult. My point stands: you need to understand this area of mathematics before you can put together the appropriate structure. There is more to this than just knowing what the various definitions are and how you can manipulate the various entities: it involves having a holistic appreciation for the entire thing. --prime mover (talk) 19:24, 22 March 2013 (UTC)


 * You can delete my posts all you want, but you'll have to admit: truth hurts.


 * I will continue to delete your schoolyard insults. You may have more mathematical experience than I do, but neither of us has learned everything there is to learn about math.


 * Oh, and chance will be a fine thing if you added any sources at all without being leaned on really heavily. My view is that the better sources are early sources. Modern textbooks are nothing more than tired regurgitation in ever more pointlessly expensive containers. It's the only way mathematicians can make money: throw together a grandiosely packaged and disgustingly expensive bag of rubbish and then make all the poor stupid undergrads buy it before they are allowed to graduate. The hope is that all one would need to do would be to read ProofWiki and not need to tap into that legalised mugging that is the "education system". --prime mover (talk) 18:45, 22 March 2013 (UTC)


 * I am not saying, by any means, that old textbooks are bad and new textbooks are good. My personal experience suggests that age is not a good predictor of quality. I am saying that in order to avoid being permanently stuck in the mathematics of the 60s, we need to take care to make room for more modern ideas and terminology. The older terminology and treatments can be put into modern perspectives. This doesn't require going and changing everything always to the most recent pronouncement of so-and-so, but it requires some willingness to rename older definitions as appropriate to allow newer ones to take the spotlight. --Dfeuer (talk) 19:08, 22 March 2013 (UTC)


 * Indeed, assessing the quality of a textbook is a hard thing to do. Invariably one needs to have multiple accounts of the same subject or bias will occur. Since is a long stretch from the subjects that are considered to have only one good source work, this shouldn't hamper things too much. In general, a book's age and number of editions/printings are indications of how good it is; another one is reviews, though it is usually hard to come by some good ones. As a final refuge, one could read the book for oneself before considering posting it up here (I highly recommend this last step be carried out to a reasonable extent for any source).


 * As for the case at hand, one definitely needs above-average erudition and subject-specific knowledge to make the mess that mathematicians have made of set theory a bit more sensible. Besides a completely rigorous account of ZF(C), which has the largest collection of authoritative resources developing it, it is IMHO best to read multiple books on the subject (not necessarily completely thoroughly; after a while one understands the most common trains of thought) but sufficiently deep as to be able to discern the idiosyncratic from the general consensus in any particular source. As-is, I fear it is necessary to conclude that not a single contributor matches these criteria (though the ubiquitous e-book versions may aid in that regard without hitting the wallet too much &mdash; consider it academic research if you worry about legal issues). I will be pleased when I hear someone feels to be up for the job. Until then, I plead it be left alone. &mdash; Lord_Farin (talk) 19:50, 22 March 2013 (UTC)


 * Left alone doesn't look very good to me. The site currently adheres more or less to the set theory enshrined in a single text, which no one appears to have, plus errors from various contributors which are rather difficult to correct given the shaky foundations. As I've said, I'm going to put up Smullyan and Fitting's version of NBG in my user space as long as I feel like it, but I do hope that it will be possible to move out of user space before ProofWiki implodes entirely. --Dfeuer (talk) 19:58, 22 March 2013 (UTC)


 * I hope so too. In fact, I hope to be back before it implodes (and to avert that event). Point is, we don't know how to implement various approaches properly right now, and it seems bad to add to the _beep_ that's already out there. I presume you refer to the Takeuti/Zaring thing with the single text; I've expressed my opinion on that one in the past. I like to think that there's not too many plainly false or severely flawed material out there. If you find a dissonance with this in your own mind it might be good to assemble a list of things that you think need to be addressed. A motivation is duly appreciated.


 * Finally, please do continue drafting pages in your user namespace about NBG theory; that way we can easily put out a lot of material relatively quickly when we get the paradigm sorted out. Investigations on further sources that could be used (if not covered fully, then at least for comparing with the Takeuti/Zaring material to assess what's particular to them, and what is considered common practice) would be most appreciated. Usual standards for bringing in sources apply. &mdash; Lord_Farin (talk) 20:17, 22 March 2013 (UTC)