Russel's paradox gives us one easily definable set that can't possibly exist because its
existence is self-contradictory.
We're now going to have a little look at the problem of sets that can't exist.
We're going to start with one very important and useful one.
The set of all sets; there is no set of all sets.
Well, why's that?
This is a real nuisance, by the way.
Supposing there were one?
Suppose that V is a set of all sets.
Well, in that case, we can apply the subset axiom scheme.
And we can define the set D
That I've written: the set of all X in Big V so that
X such as X is not a member of itself.
But because V is a set of all sets, D is simply the set of all sets that are not members of
themselves.
It's a Russel's paradox set.
And we've already seen that that set
Can't exist.
So we've got a contradiction.
So there can't, in fact, be a set of all sets.
As I say, that's a nuisance.
Yeah, there are lots of other sets, useful sets that don't exist. I've listed a few.
The set of all one elements sets, the set of all cardinal numbers, the set of all ordinal numbers.
There are lots.
And from time to time, it really gets to us.
So we can't talk about these things as being sets.
So we have a workaround.
There is an intuition for which things,
which things with curly brackets around them can be sets and which can't by the way.
The things that can't be sets are all in a particular sense too large.
They've got too many elements. They are just too big.
The proof of that statement is not on the syllabus and involves the axiom replacement.
Anyway, so sets that don't exist.
We really want to be allowed to talk about them.
So what do we do?
Well, we.
We do an abusive notation, which is what we do.
Anything we can write down with curly brackets around it
we refer to as a class.
So I've written it.
The set of all X satisfying some property Phi
So classes fall into two lumps.
A class which whose existence we can prove.
From the axioms of set theory is a set and the others are called proper classes.
Using the axiom of pairs, we can,
It's possible to show that a class is a set
if and only if it's an element of another class.
That's a criterion
you often see quoted.
OK, so I'm going to allow myself to talk about proper classes in this lecture course, because
it's just so useful to be able to do so.
So I'm going to talk about the class of all sets, the class of all cardinal numbers.
The class of all ordinal numbers and so on.
Now.
One might regard this as all a bit dodgy.
One might, might regard it as being very doubtful to refer to things that don't exist.
However, any mentioned classes can actually be got rid of.
If you care sufficiently. I've given an example.
So instead of saying Kappa is an element of the class of cardinal numbers, we can say
Kappa is a cardinal number
Erm...
When we get on to defining operators on proper classes, instead of saying, instead of saying
yeah, this is an operator on the class of all ordinals, we can say F of Alpha is defined whenever
Alpha is an ordinal or something like that.
We can always get around talking about proper classes if we care enough.
And so the fact that we can always get rid of mention of proper classes turns out to
be permission to use them.