When is a collection of sets too large to be a set?

[GargleBlast42]

Is there any easy way to say when a collection of sets is too big to be a set? For example, why is the collection of all groups, vector spaces, etc. not a set anymore? How do I determine that a given collection is still a set?

 

[Hurkyl]

One good way to show a collection X is a proper class is to exhibit an surjection X --> Set to the class of all sets. Or, similarly, an injection Set --> X.

Sometimes, you can directly translate Russell's paradox or other diagonal arguments to apply to X


To show something is a set, the easiest way is usually to show it's a subclass of something you know is a set.

 

[Martin Rattigan]

ZFC seems to be the preferred basis for mathematics. How would that work in ZFC? I.e. how would you exhibit your surjection or injection. How would you refer to the class of all groups or all sets?

 

[Hurkyl]

By it's graph, as usual. For example, the relation P(X,Y) defined by

P(X,Y) := (Y is a group) and (Y is the^* free group on X)​

is an injective function from Set to Grp.

Set is, of course, the collection of all things in ZFC. Grp is the collection of all things that are groups. Both are easily definable by predicate in ZFC.


Incidentally, when dealing with proper classes, NBG set theory is essentially the same as ZFC, but is more convenient, since it allows us us treat classes as objects rather than as first-order logical predicates.

Even more convenient is to adopt a large cardinal axiom, although doing so really is a stronger assumption than merely assuming ZFC.


*: Normally we only care about free groups up to isomorphism -- but here, for simplicity, I will suppose we have fixed a particular construction of the free group on a set[/size]