Beyond infinity.
K. Buzzard
Nov 2012, Maths in Action
Kevin Buzzard
Beyond Infinity
A typical A-level student sometimes runs into the notion of
infinity. Here’s an example that some of you may have seen:
∞
X
1
1 1 1
= + + + · · · = 1.
2n
2 4 8
n=1
Two infinities in the above line:
1) An “explicit” infinity (the ∞ symbol)
2) An “implicit” one (the “· · · ”).
Are we absolutely sure that we know what they mean?
Is ∞ even a number?? Or is it a convention? Or something
else?
Kevin Buzzard
Beyond Infinity
In this talk, a “number” will just mean a positive whole number –
like 1, 2, 3. . . .
When I talk about the “number line”, I mean the number line that
you first learn about when you are about 6 years old; it starts at
1 and only has whole numbers on it: it goes 1, 2, 3, 4, 5,. . . .
There is a really really important thing that I need to tell you
about numbers.
Kevin Buzzard
Beyond Infinity
Infinity is not a number.
Infinity definitely is not a
number.
INFINITY IS NOT A
NUMBER.
Kevin Buzzard
Beyond Infinity
Where would infinity be on the number line? “At the end”?
There is no end. Given any number, we can always add one.
Let’s get this straight – what does this idea actually prove?
It proves that there are infinitely many numbers! But every
single one of them is finite.
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
If you try to use infinity in the same way as you use numbers,
then loads of things you want to be true don’t work any more.
For example
∞+1=∞
Now take away infinity from both sides and deduce
1 = 0.
Not good.
Easy to make up other examples.
Kevin Buzzard
Beyond Infinity
So how can we possibly do mathematics with infinity?
Let’s get back to the one sensible statement I’ve made about
infinity – “there are infinitely many numbers”.
Idea: why don’t we stop thinking about numbers and start
thinking about sets. There isn’t an infinite number – but there is
an infinite set!
Kevin Buzzard
Beyond Infinity
The plan:
1) Use sets to represent a “generalised” notion of number.
2) Come up with a way of saying that two sets have the same
size.
3) Represent a usual number like 3 by a set with three
elements.
4) Represent ∞ by a set with infinitely many elements (for
example the set of all (positive whole) numbers).
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
The notation mathematicians use for sets is this:
X = {a, b, c}.
A set is just a “collection of stuff”.
X is not a number! It’s a set.
There is a number associated with X though – its size. X has 3
elements.
Kevin Buzzard
Beyond Infinity
Here’s another set!
N = {1, 2, 3, 4, 5, . . .}
N is the set of all positive whole numbers.
N is not a number – it’s a set. And N has infinite size.
I am not saying “N = ∞” here – just like I wasn’t saying “X = 3”
before.
All I’m saying is that X = {a, b, c} has three elements, and N
has infinitely many.
Kevin Buzzard
Beyond Infinity
So instead of thinking about numbers, we can think about sets,
and then we do have a working notion of infinity.
But here’s a problem: different sets can have the same size!
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
We have “too many threes” now:
X = {a, b, c}
and
Y = {p, q, r }
both have size 3, but they are not the same set.
X and Y are not the same, but they have the same size.
How can we tell if two sets have the same size??
Kevin Buzzard
Beyond Infinity
Two sets have the same size if we can “match up their
elements.”
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
Let A and B be sets.
A function f from A to B is a rule. You feed in an element of A,
and you get out an element of B.
A bijection is a special kind of function from A to B: it’s a
function with the following property:
Every element b of B is equal to f (a) for a unique element
a ∈ A.
Much easier to understand in pictures:
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
We say two sets have the same size if there’s a bijection
between them. We also say the sets biject with each other.
We need to introduce this notion, to make sure we have one,
unambiguous, interpretation of a number like 3.
A set has three elements if, and only if, there’s a bijection
between it and the set {a, b, c}.
We could even define three as “the collection of all sets that
biject with {a, b, c}.
Kevin Buzzard
Beyond Infinity
So perhaps we could define infinity as “the collection of all sets
that biject with N = {1, 2, 3, 4, 5, 6, . . .}.
Let me go through this again. Listen carefully to this bit,
because it’s subtle.
Let’s divide the collection of all sets into “classes”.
Two sets are going to be in the same “class” if they biject with
each other.
Let’s go back to our set X = {a, b, c}. Which sets are in the
same “class” as X ?
It’s the sets that biject with X .
So it’s the sets with three elements!
So if we divide the collection of all sets into classes in this way,
there is exactly one class that contains all the sets with three
elements.
Let’s call that class “3”.
Kevin Buzzard
Beyond Infinity
All our numbers: 1, 2, 3 and so on, have a unique class
associated with them. The class associated to 3 is the class
containing all sets with three elements.
Let me remind you: two sets are in the same class if they biject
with each other. That’s what it means to be in the same class.
OK so let’s define infinity to be the class containing the set
N = {1, 2, 3, . . .}.
Infinity: the collection of all sets that biject with N.
I guess every infinite set is in that class.
Actually. . . hmm. . . let’s check some examples, just to make
sure.
Kevin Buzzard
Beyond Infinity
Let’s define E = {2, 4, 6, 8, . . . , }, the positive even numbers.
Can we find a bijection between E, the positive even numbers,
and N = {1, 2, 3, . . .}? Are they in the same class?
Or is E really “a smaller infinity” than N? Is there more than one
class of infinite sets? Is that a crazy idea?
Kevin Buzzard
Beyond Infinity
E and N are the same size: there’s a bijection between them.
The sets E and N are in the same class.
Here’s the bijection. Define f : N → E by
f (n) = 2n.
So f (1) = 2, f (2) = 4, f (3) = 6,. . .
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
OK, well, what about Z, the set
{. . . , −3, −2, −1, 0, 1, 2, 3, . . .}
of all integers? Does Z biject with N? Is it in the same class?
Or is Z a bigger infinity than N?
Z and N biject with each other. They’re in the same class.
Kevin Buzzard
Beyond Infinity
Kevin Buzzard
Beyond Infinity
OK so what about R, the real numbers?
Theorem (Cantor)
R does not biject with N.
The set of reals is a “bigger” infinite set than the set of positive
whole numbers!
If we put sets into classes, by saying two sets are in the same
class if they biject with each other – then there is more than
one class of infinite sets.
Hence there is more than one kind of infinity.
In fact there are infinitely many infinities!
For example, if W denote the set of all sets of real numbers(!),
then the size of W an even bigger infinity than the size of R.
Kevin Buzzard
Beyond Infinity
Here is a proof of Cantor’s theorem that there is no bijection
from N to R.
Let’s consider any old function f : N → R.
For example, it might look like this:
f (1) = 5.75347563476537845 . . .
f (2) = 63847.34584385638476 . . .
f (3) = 3
f (4) = 123.45678
and so on.
Kevin Buzzard
Beyond Infinity
f (1) = 5.75347563476537845 . . .
f (2) = 63847.34584385638476 . . .
f (3) = 3.000000000 . . .
f (4) = 123.4567800000 . . .
The red number in f (1) is the first digit after the decimal point.
The red number in f (2) is the second digit after the decimal
point.
Let’s define a new real number r by making sure that the nth
digit in the decimal expansion of r is definitely not the same as
the nth digit in f (n).
In our example, r could be 0.8518 . . ..
Then r cannot be f (n) for any n!
Hence f is not a bijection. QED.
Kevin Buzzard
Beyond Infinity
Conclusion: There is no bijection from N to R!
In other words: N and R are in different classes.
The size of N is “the smallest infinity” (this is not too hard to
make precise).
Is the size of R “the second smallest infinity”?
Kevin Buzzard
Beyond Infinity
The continuum hypothesis is the statement that R is the
second-smallest infinity.
Here’s how to make that assertion rigorous.
Hypothesis (Continuum hypothesis)
If X is a set of real numbers, then one of the following holds:
(1) X is finite
(2) X bijects with N
(3) X bijects with R
For example, the set of even numbers bijects with N, as does
the set of all integers.
And the set of positive reals bijects with R (use log).
The continuum hypothesis says that if we look at any set of real
numbers, it’s either in a finite class, or in N’s class, or in R’s
class.
Is the continuum hypothesis true though?
Kevin Buzzard
Beyond Infinity
Theorem (Gödel, Cohen)
The continuum hypothesis is independent of mathematics.
In other words, there are some models of mathematics where it
is true, and others where it is false.
[Strictly speaking there is a third possibility – that there are no
models of mathematics at all! But let’s not go there.]
Kevin Buzzard
Beyond Infinity
What you probably think of as “mathematics” is one abstract
gadget with sets, numbers, functions, and where everything is
either true or false.
That’s what everyone thought until around 100 years ago.
But here’s the truth. Mathematics is any abstract gadget which
satisfies a few simple rules.
The questions you get asked at school are questions which
have only one answer, because you can deduce the answer
from the simple rules.
But there are actually infinitely many models of mathematics.
And when you’re all doing your PhD’s and get to the boundary
of what is known, you may run into questions whose answer
depends on which model you’re using.
[the end]
Kevin Buzzard
Beyond Infinity