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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)


 * Fair enough, so the job's too challenging for you. So rather than do a half-assed version, it would be better to leave it for someone else to take on board. Work on something else instead. --prime mover (talk) 06:14, 22 March 2013 (UTC)