Definition talk:Small Class

I can't get the difference between:

1. Small class
2. Set variable
3. Set

--prime mover (talk) 21:24, 20 August 2012 (UTC)

A small class is a definitional abbreviation for (ultimately) a property which' extent (i.e., the collection of sets satisfying it) is a set. A set variable is a variable in the language of set theory. A set is an element of a model for the language of set theory.

Small classes are identified iff they have the same extent, so that they may also be viewed conceptually as '(things effectively equal to) sets'. Proper classes are 'things which look like sets but are external to the model' (which is why they can't consistently be called 'sets'). I hope to have contributed to enlightenment. --Lord_Farin (talk) 21:32, 20 August 2012 (UTC)

What's the difference between a "thing effectively equal to a set" and "a set"? What does "effectively" mean? What is the difference between a small class and a set?

As for "set variable", I understand that "Language of Set Theory" is a formal system which can be interpreted as a model for the way sets work. Therefore a set variable is not a set, but in any given LaST theorem it can be considered as being replaceable with a set, in that if all set variables were replaced by sets, the given statement would be a true statement in set theory. Or something.

Right? Wrong? Overcomplicated? Oversimplified? Cutting through the bullshit? --prime mover (talk) 22:17, 20 August 2012 (UTC)

The notion of "sets" was already here when I got here. They are treated essentially the same as small classes, except that small classes are often proven to be small and there are theorems surrounding whether a certain class is indeed small (whereas the treatment of "sets" here is a little more naive). So if they are something, sets are small classes. The main difference is the context in which they are used: small class is usually used here when ZF set theory is invoked specifically, whereas sets is used when the result is not considered a "set theory" one and whether "existence" of the set is not an issue. --Andrew Salmon (talk) 00:50, 21 August 2012 (UTC)

So a set is a small class is a set, then? --prime mover (talk) 05:16, 21 August 2012 (UTC)

Yes. Just that they're used in different contexts here. --Andrew Salmon (talk) 05:34, 21 August 2012 (UTC)

In which case would it not be appropriate to cut through the confusing language "small classes correspond to sets" and tell it like it is: "Small classes are sets"? --prime mover (talk) 05:57, 21 August 2012 (UTC)

As I tried to point out: Sets and small classes are intuitively describing the same thing, but the former is the internal notion to the model of set theory, and the latter is the (an) external notion. I contend that there is value in this distinction because in other situations the internal and external notions may not be equivalent (in that theorems for the one not necessarily hold for the other). It's complicated, but that is how I view it. --Lord_Farin (talk) 06:02, 21 August 2012 (UTC)

As you say, it's complicated. This distinction needs to be explained formally on the page itself, then. The current exposition uses impenetrable language which obscures rather than enhances understanding. Sorry to go on and on about this, but if the distinction is important (as is clear) then this needs to be explained properly. And it's currently not being. --prime mover (talk) 06:11, 21 August 2012 (UTC)

Would you consider the new attempt an improvement? --Lord_Farin (talk) 06:22, 21 August 2012 (UTC)

It's definitely better. I'm not sure but it may still have room for improvement, though. --prime mover (talk) 06:26, 21 August 2012 (UTC)

---

There is an added nuance to this.

In the analysis as performed by 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.), all classes are considered as subclasses of a basic universe, which itself is subset to those "incontrovertible" and "obvious" axioms common to ZFC and NBG and (probably) all such models: axioms of: empty set, unions, pairing and powers. A basic universe is also subject to two additional axioms: the Axiom:Axiom of Transitivity and Axiom:Axiom of Swelledness (the latter up for grabs for anyone with a less kindergartenesque name to coin). Right at the top (or at the bottom, depending on whether your approach is sephirothic or qliphothic) we have the Axiom:Axiom of Extensionality and Axiom:Axiom of Specification (the latter of which has many names, but at root allows you to subclass).

Now it just so happens that one of the direct interpretations of the Axiom:Axiom of Transitivity is that "every set is a class".

So, accepting the notion of a basic universe which is "the class of all sets" subject to the above axioms, the concept of a "small class" is no longer an issue. A set is a class, and so the idea of "needing" to establish a class which is "equal to" (but not "the same thing as") a set is no longer there.

At some stage we may need to take another deep long philosophical think about the nature of "equality", and wonder whether Leibniz's law may itself need to be scrutinised. --prime mover (talk) 10:36, 21 May 2022 (UTC)

Well, every set may be a class, but certainly not every class is a set. And while I agree that we could rewrite all proofs of "The class of X is small" into "The class of X is a set" this does not alter the meaning that much. In fact that is the only direction (formalities and proof techniques aside) that is relevant in doing actual mathematics.

The essential difference is that without proving a class small, we are not permitted to quantify over it. It is just a "notational device". But when it is small, we can do all the set thingies with it and a world of possibilities opens.

For me personally, the merit of "small class" lies in that it reminds us that there is something nontrivial about it; that we have taken into account Russell and his associates and still formally conclude our result. Whereas "Class of X is a set" does not express that awareness as clearly. — Lord_Farin (talk) 08:37, 22 May 2022 (UTC)

Sorry L_F, I'm afraid I absolutely do not get this concept at all. --prime mover (talk) 08:40, 22 May 2022 (UTC)