Definition talk:Class (Class Theory)

Classes vs. Sets

It seems like a lot of potential ambiguity / paradoxes arise from a lack of verbal syntax as far as referring to classes vs. sets. From how I understand it, when we talk about sets, we are only about elements of the universe of discourse, no? And classes are collections of these sets. However, for example, the definition of a singleton refers to them as "sets" which seems to presuppose the existence of sets that contain exactly one element (which requires the axiom of pairing or axiom of power set to prove in reality). -Andrew Salmon 17:28, 12 September 2011 (CDT)

What's your point? --prime mover 00:37, 13 September 2011 (CDT)

Only when things have been proven to be elements of the universe of discourse should they be considered and referred to as sets...this, however, is easy to tie up. -Andrew Salmon 00:43, 13 September 2011 (CDT)

Removed Comment

I have removed the comment about ZF and Von Neumann-Gödel-Bernays set theory, because ZF does allow the use of classes. The difference between the two systems is that you can quantify over classes in NGB, but you cannot in ZF. --Andrew Salmon 03:37, 7 August 2012 (UTC)

What comment was that? I can find no evidence of its existence. --prime mover 13:52, 7 August 2012 (UTC)

http://www.proofwiki.org/w/index.php?title=Definition%3AClass&action=historysubmit&diff=101294&oldid=62162, I think. --Lord_Farin 13:55, 7 August 2012 (UTC)

Is a class a set?

"A class is small if it is equal to some set". Does this mean "A small class is a set", that is, "a set is a small class"? Is there a subtlety here such that a "small class" is not actually a set but just "equal to it"? Because all this seems circular: there's a page called "All Sets are Small" (better, as LF says, to rename it "Set is Small Class") and then when you link to "small class" it leads you to a page that says "a small class is a set". How does this work? --prime mover 13:58, 7 August 2012 (UTC)

There is something to be said here, which has to do with the intricate concepts 'internal' and 'external'. In the current reference, we call a 'set' an element of the particular model for the theory. As such the language of the theory can say things about it; hence it is called 'internal'. A 'class' on the other hand is 'external', something the 'residents', so to speak, of the model can't reason about; it is outside their realm of comprehension (similar to the flatlanders being unable to grasp the third dimension). We however, being external to the theory, can talk about 'classes'. Certain external 'classes' have an internal 'set' counterpart (namely, the so-called 'small classes'). Our external notion of 'set' is used here in a disputable way, in that we make reference to the ZF axioms and Russell's paradox to think about what is, and what isn't to be called a 'set' (externally). IMO it would be best to avoid the words 'is' and 'equals' when comparing internal and external stuff; rather, 'corresponds' or something similar would best be used. In this sense, you were spot-on with your suspicion that there was an intricate 'equal to, but not actually the same'-thing going on.

We are here still in the pleasant realm where internal and external things do not collide. But it may well be (forgive me my topos theory dictionary) that we have, in certain cases, that a "topos" satisfies classical logic in an 'external' sense, but e.g. modal logic or intuitionistic logic for 'internal' purposes. It's just that some things can't be expressed in the language of the theory just as well as in natural language. This stuff messes with one's mind, and my mind has become very messy the last few weeks :). More into the standard realm, we have theorems asserting a countable model for set theory exists. However, there will be 'uncountable' sets in that model. We see that the internal expressivity of the language apparently requires that the internal definition of 'countable set' is different to the external one. I hope I have conveyed some of the rationale and principles behind all this confusion.--Lord_Farin 14:20, 7 August 2012 (UTC)

To this end, I believe it important, when extracting a package of material from a book discussing axiomatics, to start from the beginning and making sure the basic definitions correspond with that in the database. Most of the books I have in this area start with something like "A set is an aggregation of elements" or some such, and then either go on from there into arithmetic / abstract algebra, or go completely the other way and say "...but because of Russell's paradox and all that, we need to refine our understanding." Once you get to chapter 4 of such a book, the objects that have been defined and explored are (without all this background) more or less unrecognisable, despite being "basically the same thing as" the material which is already there.

In the contexts of these more abstract modern developments, all the things we take for granted like e.g. "equality" can't be trusted to mean what we think they ought to mean, and so it is probably a good idea, rather than overload the pages containing the simple concepts with "abstract nonsense", to create completely separate pages within those more modern contexts, which can be accessed as needed by pages which need those more precise definitions. For normal use, "equals" means "is the same thing as" and can be taken for granted as meaning just that (you don't need all the heavy machinery of class theory when working in elementary calculus, for example), but if there is a new, alien viewpoint (like this one of "class" versus "set" as LF describes above) then the explanatory material needs to be written, and written rigorously. --prime mover 14:59, 7 August 2012 (UTC)

En passant, you appear now convinced that there is a cause for duplicating the set theory results to classes. A good thing, given the rigour we strive for. I think we should be quite pedantic in adding references like 'This result generalizes "Result in set theory"' in the Also see section to maintain the desired interlinking; not sure that links the other way round are useful, though... I would start contributing in this area, but I have no physical, authoritative sources on this subject, and my eBook collection is up for a repair with the rest of my desktop; I will thus keep it to buzzing around Asalmon's head and through his contributions, enforcing and promoting rigour where I can (whereof the several discussions yesterday and today were a product). --Lord_Farin 15:11, 7 August 2012 (UTC)