Thursday, March 23, 2006

Sets, Classes and Voodoo

After much anticipation I started reading Barwise and Moss's "Vicious Circles". Unfortunately by chapter 2 I've reached the point where the statements don't just seem incorrect, they don't even seem to be propositions. I raised my question on USENET but I may as well mention it here too.

Here's how I understand the concept of a class in ZF Set Theory: talk about classes is really just talk about predicates. We enrich the language of Set Theory with a bunch of new terms ('class', 'subclass') and overload other terms ('is an element of', 'is a subset of') to give a new language that reifies classes, but instead of adding new axioms to deal with classes we provide a translation back to ZF without these extra terms. For example if P and Q are classes then "x is in P" means "P(x)" and "P is contained in Q means" "for all x, P(x) implies Q(x)" or even "a subclass of a set is a set" which translates to the axiom of separation. (We could alternatively add new axioms, instead of the translation, and then we'd get NBG Set Theory.)

Am I right so far? (By the way, nobody ever seems to say what I've just said explicitly. In particular, it seems to me that once you add the term 'class' you need to start proving metatheorems about classes to show what kind of deductions about them are valid, but nobody ever seems to do this.)

I understand that this is a sensible thing to do because of the overloading - in the enriched language sets and classes look similar and that allows us to do category theory, for example, in a much wider context, without having to define everything twice. (And also, maybe, because talk about sets is really a substitute for talk about classes...but that's a philosophical point for another day...)

So what does "If a class is a member of a class then it is a set" mean in the context of ZF? A class is really just a predicate, so it doesn't make sense to me that there could be a predicate about predicates. So at this point the book is looking like Voodoo.

Can anyone out there clarify this for me?

(Hmmm...I wonder if the translation to ZF is an adjoint functor...ow...MUST STOP OBSESSING ABOUT ADJUNCTIONS...)

5 comments:

David R. MacIver said...

I'm not sure that no one does that explicitly. Certainly the course on set theory I had from Imre Leader did. I don't know about textbooks though...

The only reasonable interpretation I can make of that sentence is that it is a very poorly stated way of saying that it doesn't make sense to talk about a class of classes. For example you might be tempted to talk about the class of categories, but oops. You can't do that, as there are (rather a lot of) categories which are proper classes.

It's not a proposition, as there's nothing to be proved. It's just an observation that we can't overload *all* our terms to apply to classes in general.

I think.

sigfpe said...

David,

You're making me feel happier. You actually know about this stuff and seem to be agreeing with me. When I check out the reviews on the web they basically seem to say that this book has great stuff in it but it's a little sloppy.

So you're a Cambridge man eh? I remember Imre though he won't remember me.

Kenny said...

The way you're talking seems to be the standard way to think about classes in ZF. You're right that it then doesn't then make sense to say that "if a class is a member of a class then it is a set". However, there are several axiomatized "class theories" that extend ZF - I think the most common is NBG (von Neumann-Bernays-Godel), and the other I've heard of is MK (Morse-Kelly). The distinctions between them are discussed briefly at the end of the first chapter of Kunen's set theory book. I think the basic idea is that we have a two-part ontology, with both sets and classes. The sets are governed by ZF, while the classes are governed by extensionality and unrestricted comprehension over sets (I think), plus the axiom that if a class is a member of something then it is a set.

I believe the standard way to make sense of all this is to take a model of ZFC with some large cardinals. Then if I is an inaccessible cardinal, then V_I is a model of ZFC, while V_{I+1} will be a model of NBG, if you let the sets be the elements of V_I and the classes be all the elements of V_{I+1}.

sigfpe said...

Thanks Kenny.

The book is explicitly talking about ZF but I can now imagine a plausible scenario by which this sentence might have found its way into this book if the authors were being a bit sloppy.

Anonymous said...

1. "If a class is a member of a class then it is a set"

2. There is a class which contains *everything*

Conclusion 1: Every class is a member of this class

Conclusion 2: Every class is a set

What the heck? Am I confusing classes with something other? Am I misunderstanding the sentence "If a class is a member of a class then it is a set"?

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