Wikipedia:Reference desk/Archives/Mathematics/2015 November 6

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November 6

Basic Set theory question: Difference between a Class and a Set?

Just to be clear in my question I am NOT talking about an object class as the term is pervasively used in computer science. I know that stuff (very well if I do say so myself). I'm trying to refresh and extend my knowledge of set theory. I've known the basics but as one of my old bosses used to say I have to "hum the math" when it comes to proofs and I want to get to the point where I can really understand the details of things like Godel's proofs and the diagonalization arguments that describe the different kinds of transfinite numbers. I'm stuck on what I think is probably a fairly simple distinction. In the book Basic Set Theory by Azriel Levy he draws the distinction between a class and a set very early on. And I've read that section several times and I'm still not quite getting it and then as I read on I keep getting hung up wondering why he needs to use class in some proofs and sets in others. Can someone give me a simple description of what the difference is? For example are all sets also classes? What can you not do with a class that you can do with a set? Or is there some other distinction I'm missing? I get the feeling the distinction comes in because of Russell's paradox and the Axiom of Comprehension (which I think was also Frege's infamous Basic Law V) let you define things like the set of all sets that aren't a member of themselves but I don't see how classes get you out of that paradox, or maybe I'm just confused about why we need them. Any guidance would be helpful. I kind of feel toward this stuff the way I do toward music, I don't have much talent for it but I love it anyway ;) --MadScientistX11 (talk) 16:05, 6 November 2015 (UTC)

The modern conception of set is the von Neumann hierarchy. You start with the empty set, you take its powerset, you take the powerset of that, you take the powerset of that, and so on. Once you've done that for all the natural numbers, you take the union of everything you've seen, and that gives you V_ω. You take the powerset of that, which gives you V_ω+1, and so on.

You keep iterating this through all the ordinal numbers.

Anything that shows up at any point, as an element of something you get to in this process, is a set.

However, the collection of all sets, V, does not ever show up in this process, and is not a set. It is a proper class.

There's more that could be said, but this is the most basic and useful starting point, I think. --Trovatore (talk) 18:29, 6 November 2015 (UTC)

I'd say that a class is a property satisfied by some (or all) sets; every set is a class, namely the class whose defining property is membership in that set; and the thing you can do with sets but not classes is make them members of sets or classes.

In ZF, the class of all sets that aren't members of themselves is the same as the class of all sets. It doesn't contain itself, but there's no contradiction because it isn't a set. -- BenRG (talk) 23:26, 7 November 2015 (UTC)

Since "every set is a class" (as you correctly claim), then you cannot claim that "a class is a property satisfied by some (or all) sets". On the contrary: being a set is a property satisfied by some classes, namely: for every class, it is a set if and only if it is a member of some class; Of course, that class - containing that set as a member - can be a set, but it is unnecessarily a set - i.e. it is unnecessarily a member of some class. HOOTmag (talk) 03:03, 8 November 2015 (UTC)

I think Ben is saying that each class is a property, not that all sets have the property of classness. For example, some sets are von Neumann ordinals, and there is a class of von Neumann ordinals, identified with the property of von-Neumann-ordinal-ness.

This is the extensional-v-intensional distinction, and it's part of the "more that could be said" bit that I left unsaid. --Trovatore (talk) 03:58, 8 November 2015 (UTC)

It seems you're right. HOOTmag (talk) 08:16, 8 November 2015 (UTC)