K. Buzzard Nov 2012, Maths in Action - Imperial College...

Beyond infinity.

K. Buzzard

Nov 2012, Maths in Action

Kevin Buzzard Beyond Infinity

A typical A-level student sometimes runs into the notion ofinfinity. Here’s an example that some of you may have seen:

∞∑n=1

12n =

12+

14+

18+ · · · = 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 somethingelse?


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 thatyou first learn about when you are about 6 years old; it starts at1 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 youabout numbers.


Infinity is not a number.

Infinity definitely is not anumber.INFINITY IS NOT ANUMBER.


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 everysingle one of them is finite.


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.


So how can we possibly do mathematics with infinity?

Let’s get back to the one sensible statement I’ve made aboutinfinity – “there are infinitely many numbers”.

Idea: why don’t we stop thinking about numbers and startthinking about sets. There isn’t an infinite number – but there isan infinite set!


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 samesize.

3) Represent a usual number like 3 by a set with threeelements.

4) Represent∞ by a set with infinitely many elements (forexample the set of all (positive whole) numbers).


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 3elements.


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 Nhas infinitely many.


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!


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??


Two sets have the same size if we can “match up theirelements.”


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 afunction with the following property:

Every element b of B is equal to f (a) for a unique elementa ∈ A.

Much easier to understand in pictures:


We say two sets have the same size if there’s a bijectionbetween 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 bijectionbetween it and the set {a,b, c}.

We could even define three as “the collection of all sets thatbiject with {a,b, c}.


So perhaps we could define infinity as “the collection of all setsthat 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 witheach other.

Let’s go back to our set X = {a,b, c}. Which sets are in thesame “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 threeelements.

Let’s call that class “3”.Kevin Buzzard Beyond Infinity

All our numbers: 1, 2, 3 and so on, have a unique classassociated with them. The class associated to 3 is the classcontaining all sets with three elements.

Let me remind you: two sets are in the same class if they bijectwith 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 setN = {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 makesure.


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 oneclass of infinite sets? Is that a crazy idea?


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,. . .


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.


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 positivewhole numbers!

If we put sets into classes, by saying two sets are in the sameclass if they biject with each other – then there is more thanone 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.


Here is a proof of Cantor’s theorem that there is no bijectionfrom 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.


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 decimalpoint.

Let’s define a new real number r by making sure that the nthdigit in the decimal expansion of r is definitely not the same asthe 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 tomake precise).

Is the size of R “the second smallest infinity”?


The continuum hypothesis is the statement that R is thesecond-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 doesthe 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 realnumbers, it’s either in a finite class, or in N’s class, or in R’sclass.Is the continuum hypothesis true though?


Theorem (Gödel, Cohen)The continuum hypothesis is independent of mathematics.

In other words, there are some models of mathematics where itis true, and others where it is false.

[Strictly speaking there is a third possibility – that there are nomodels of mathematics at all! But let’s not go there.]


What you probably think of as “mathematics” is one abstractgadget with sets, numbers, functions, and where everything iseither true or false.

That’s what everyone thought until around 100 years ago.

But here’s the truth. Mathematics is any abstract gadget whichsatisfies a few simple rules.

The questions you get asked at school are questions whichhave only one answer, because you can deduce the answerfrom 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 boundaryof what is known, you may run into questions whose answerdepends on which model you’re using.

[the end]Kevin Buzzard Beyond Infinity

K. Buzzard Nov 2012, Maths in Action - Imperial College...

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