When people started building mathematics on set theory and on set theory and logic,
they were very optimistic.
So you may have heard of Russell and Whitehead's Principia Mathematica, you will have heard
of.
You may have heard of Frege's Foundations of Arithmetic.
There was an idea floating around that maybe you could reduce mathematics to pure logic.
You could reduce set theory to pure logic.
You could write down a list of axioms that were self-evidently just logically true and
deduce everything from them.
It was that Russell made the unpleasant discovery that that wasn't true.
So this is Russell's paradox.
What does Russell's paradox say?
Well, the background to it is that set theory is about building sets.
So we have the set of all real numbers, the set of continuous, complex, valued functions,
set of all square-integral functions.
If you've got a mathematical property φ, such as being a real number we want to look
We want to be able to talk about the set of all things satisfying that property, the
set of all real numbers and so forth.
Well, why not?
Why don't we just go ahead and do this?
Well.
This is the set that gave Bertrand Russell a lot of sleepless nights.
About 120 years ago.
Let us consider the set of all sets that are not members of themselves.
So Capital D as set all x.
Such that x is not an element of x.
So the set of natural numbers is an example, the set of natural numbers is not a natural
number.
The set of square integrable functions is not a square integrable function.
OK.
Is D a member of itself?
Well, D is a member of D.
If and only if it satisfies the criterion for membership.
And what is the criterion for membership?
Well, that D should not belong to D. Oh Dear!
D is a member of itself, if and only if it's not.
We have a logical contradiction.
The existence of D is logically inconsistent.
Okay, what do we do now?
Well, the answer is we be cautious.
We are, we have to be cautious about when we allow ourselves to say that a set exists.
And that is a very big deal.
OK, so what we're going to do is we're going to have a list of axioms, most of which will
say when we allow ourselves to think of a set as existing.
And this set is going to be and it's going to have to satisfy two opposing requirements
is going to need to be powerful so we can prove lots of stuff,
but not too powerful.
so we don't get to prove zero equals one or the moon is made of green cheese or something
like that.
Okay.
There is,
Okay.
So we we write down a set of axioms.
We're going to be using the ZFC axioms, which are one of the standard sets.
There's an additional problem.
The additional problem is that we'd really like to be sure that our, our cautious set
of axioms really is logically consistent.
Well, what's wrong?
What might go wrong with that?
Well, this is nasty surprise
number two for set theorists: Gödel's second incompleteness theorem, not on the syllabus
of this course.
It tells us - its implication for set theory is that if we have a nice set of axioms
that we want to use for set theory
We'd like to prove it is consistent, but we can only prove that it's consistent
if it isn't
Point being, of course, that from an inconsistent set of axioms, you can prove everything so
if theta is inconsistent,
you can prove everything.
You can prove in particular that ϴ is consistent.
You can also prove that I am the Pope.
And, and the moon is made of green cheese and stuff like that.
So the set, if our set theoretic axioms are logically consistent,
then we'll never be sure about it because the s second incompleteness theorem prevents
us from knowing.
So the ZFC axioms, they've existed for about 110 years now.
Nobody has yet found,
Deduced a contradiction from them
So we are optimistic, but we'll never be sure that they're okay.
That is a bit of a problem.
So when we're founding mathematics, we want to build mathematics on a foundation that's firm
and solid,
Gödel says that we'll never be sure that set theory is firm and solid.
Oh, well.
OK.
So.
So that's a background to these ZFC axioms I'm going to be presenting. I'll present them
as I need them.
And that will be intended partly to make the point that we are being really cautious, not
introducing axioms.
till we do need them.
And I'll also say from time to time, which of the axioms might pose the biggest problem
for logical consistency.
So if ZFC does get proved to be inconsistent in 2050 or whenever, I'll indicate
which of the axioms it might be that has blown our set theory out of the water.