Are Some Infinities Bigger Than Others?
Infinity is already strange. Mathematics makes it stranger: some infinities really are bigger than others.
Most students encounter infinity long before they are ever asked what it actually means.
A number line apparently continues for ever. There are infinitely many whole numbers. A graph may approach a line indefinitely. Recurring decimals continue without stopping. We sometimes even write the infinity symbol, ∞, as though infinity were simply an extraordinarily large number.
But infinity is not just a very big number.
And one of the most remarkable discoveries in mathematics is that not all infinite collections are the same size.
That sounds impossible at first. Surely once something is infinite, it is simply infinite?
Georg Cantor showed otherwise.
There are infinitely many whole numbers.
There are infinitely many fractions.
There are infinitely many real numbers.
But while the first two infinities can be matched with one another, the infinity of the real numbers is fundamentally larger.
The journey towards understanding why can begin somewhere much less intimidating than advanced set theory.
It can begin in a rather peculiar hotel.
Welcome to Hilbert's Hotel
Imagine a hotel with rooms numbered:
1, 2, 3, 4, 5, 6, ...
and suppose that the rooms continue for ever.
There is no final room.
Now imagine that every room is occupied.
In an ordinary hotel, the manager would put up the familiar sign:
NO VACANCIES
But Hilbert's Hotel is not an ordinary hotel.
Late one evening, another traveller arrives and asks for a room.
The hotel is full.
Can the manager accommodate the new guest?
Surprisingly:
Yes.
The manager simply asks:
- the guest in room 1 to move to room 2;
- the guest in room 2 to move to room 3;
- the guest in room 3 to move to room 4;
- and in general, the guest in room n to move to room n + 1.
Every existing guest still has a room.
But room 1 is now empty.
The new guest moves in.
The hotel was completely full, yet somehow had space for another person.
Welcome to infinity.
Now Send 1,000 New Guests
Suppose a coach arrives carrying 1,000 people.
Still no problem.
Move every existing guest from:
room n to room n + 1000.
Rooms 1 to 1000 become empty.
All 1,000 new guests can be accommodated.
Again, the hotel was full before they arrived.
What If Infinitely Many Guests Arrive?
Now the situation becomes much stranger.
An infinitely long coach arrives carrying guests numbered:
1, 2, 3, 4, 5, ...
There are infinitely many new guests.
Surely even Hilbert's Hotel cannot deal with that.
But it can.
Ask every existing guest to move from:
room n to room 2n.
So:
1 goes to 2
2 goes to 4
3 goes to 6
4 goes to 8
and so on.
Every existing guest moves into an even-numbered room.
That leaves every odd-numbered room empty:
1, 3, 5, 7, 9, ...
There are infinitely many of them.
Guest 1 from the coach gets room 1.
Guest 2 gets room 3.
Guest 3 gets room 5.
And so on.
An infinitely full hotel has just accommodated another infinity of guests.
That is the point at which our ordinary intuition about size begins to fail.
Infinity + 1 = Infinity?
In ordinary arithmetic:
10 + 1 > 10
and:
1,000,000 + 1 > 1,000,000.
Adding something makes a quantity larger.
But when we are discussing the size of an infinite set, things can behave differently.
The counting numbers are:
1, 2, 3, 4, 5, ...
If we add one extra object to this infinite collection, we can still pair everything with the counting numbers.
In terms of cardinality:
infinity + 1 = infinity
But this needs an important qualification.
We are not treating infinity as an ordinary number and doing ordinary arithmetic. We are talking about the cardinality, or size, of infinite sets.
That distinction becomes extremely important.
What Does "The Same Size" Mean?
With finite sets, comparing size is easy.
Suppose I have five pencils and five students.
I can give one pencil to each student.
If every student receives exactly one pencil and there are no pencils left over, the two sets contain the same number of objects.
Mathematicians call this a one-to-one correspondence.
Cantor realised that the same principle could be used with infinite collections.
Consider:
Counting numbers:
1, 2, 3, 4, 5, ...
Even numbers:
2, 4, 6, 8, 10, ...
It seems obvious that there should be fewer even numbers. After all, the even numbers are only part of the counting numbers.
Yet pair them:
1 <-> 2
2 <-> 4
3 <-> 6
4 <-> 8
5 <-> 10
and in general:
n <-> 2n
Every counting number has exactly one even-number partner.
Every even number has exactly one counting-number partner.
So the two sets have the same cardinality.
This is one of the first genuinely surprising properties of infinite sets:
An infinite set can have the same size as one of its proper subsets.
That cannot happen with a finite set.
There Are as Many Even Numbers as Whole Numbers
This makes a wonderful discussion question for students.
Ask:
Which are there more of: counting numbers or even numbers?
Most will initially say counting numbers.
That is entirely reasonable because the even numbers are contained within them.
But mathematically they are the same size.
This type of infinity is called countably infinite.
The standard mathematical symbol for its cardinality is:
aleph_0
pronounced "aleph-null" or "aleph-zero".
So:
|N| = aleph_0
where N represents the natural or counting numbers.
The notation looks advanced, but the idea behind it is surprisingly simple.
If we can arrange every member of a set into a list:
first, second, third, fourth, ...
without missing anything, then the set is countable.
What About Negative Numbers?
Now consider all the integers:
..., -3, -2, -1, 0, 1, 2, 3, ...
Surely there must be more integers than counting numbers because we have added zero and all the negative numbers.
Yet we can arrange them:
0, 1, -1, 2, -2, 3, -3, 4, -4, ...
Now give them positions:
1 -> 0
2 -> 1
3 -> -1
4 -> 2
5 -> -2
and so on.
Every integer eventually appears.
Therefore the integers are also countably infinite.
So:
|Z| = |N|
even though N is contained within Z.
Infinity is behaving strangely again.
Surely There Must Be More Fractions?
This is where the subject becomes particularly interesting.
Between 0 and 1 alone there are infinitely many fractions:
1/2
1/3
2/3
1/4
3/4
1/5
and so on.
In fact, between any two different numbers there are infinitely many fractions.
Between 1 and 2 we can find:
3/2.
Between 1 and 3/2 we can find:
5/4.
We can continue for ever.
So it seems almost impossible that fractions could be counted.
Yet they can.
Putting the Fractions Into a Grid
Imagine constructing a table.
Across the top write the denominators:
1, 2, 3, 4, 5, ...
Down the side write the numerators:
1, 2, 3, 4, 5, ...
The grid contains fractions such as:
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
and so on.
Now travel diagonally through the grid.
You might encounter:
1/1
1/2, 2/1
3/1, 2/2, 1/3
1/4, 2/3, 3/2, 4/1
and continue indefinitely.
There will be duplicates:
1/2 = 2/4 = 3/6
so we simply skip any fraction that has already appeared.
Eventually every positive fraction will appear somewhere in our list.
Negative fractions can then be incorporated in a similar way to the negative integers.
The result is astonishing:
The rational numbers are countably infinite.
In symbols:
|Q| = |N|
There are, in this very precise mathematical sense, exactly as many fractions as counting numbers.
This Is Where I Think Infinity Becomes Really Fascinating
This is one of those topics I particularly like because mathematical intuition keeps giving perfectly sensible answers — and mathematics keeps showing us something different.
If I asked a student which collection appeared larger:
- 1, 2, 3, 4, ...
- every possible fraction
I would expect most people to choose the fractions.
There seem to be fractions everywhere.
Yet the remarkable thing is that they can still be put into a list.
That word list is crucial.
If every item can eventually be assigned a position:
1st, 2nd, 3rd, 4th, ...
then the set is countable.
The obvious next question is therefore:
Can everything be counted like this?
Cantor's answer was no.
And that changed mathematics.
Enter the Real Numbers
The real numbers contain all the numbers on the ordinary number line.
They include:
- integers;
- fractions;
- terminating decimals;
- recurring decimals;
- irrational numbers such as pi;
- square roots such as sqrt(2);
- numbers whose decimal expansions continue for ever without repeating.
Now consider just the real numbers between 0 and 1.
That sounds like a very small part of the entire number line.
Surely if all the fractions can be counted, these numbers can be counted too?
Cantor proved that they cannot.
Cantor's Diagonal Argument
This is one of the most beautiful proofs in mathematics because its basic idea can be understood without university-level mathematics.
Suppose, for the sake of argument, that we could list every real number between 0 and 1.
Perhaps the list begins:
1: 0.314159...
2: 0.500000...
3: 0.271828...
4: 0.123456...
5: 0.707106...
...
We claim that this list contains every possible real number between 0 and 1.
Cantor now performs a clever trick.
Look at:
- the first decimal digit of the first number;
- the second decimal digit of the second number;
- the third decimal digit of the third number;
- the fourth decimal digit of the fourth number;
- and so on.
These form a diagonal through our supposedly complete list.
Now construct a new number.
For every diagonal digit:
- if the digit is 1, make the new digit 2;
- otherwise make the new digit 1.
The resulting decimal differs from:
- number 1 in its first decimal place;
- number 2 in its second decimal place;
- number 3 in its third decimal place;
- number 4 in its fourth decimal place;
and so on.
Therefore the new number cannot equal any number on the list.
But it is a perfectly valid real number between 0 and 1.
We claimed our list contained every real number.
Yet we have just constructed one that is missing.
Contradiction.
Therefore:
No complete list of the real numbers between 0 and 1 can exist.
The real numbers are uncountable.
Some Infinities Really Are Bigger Than Others
We have now discovered two fundamentally different types of infinity.
The counting numbers, integers and fractions are countably infinite:
|N| = |Z| = |Q| = aleph_0
But:
|R| > aleph_0
The real numbers form a larger infinity.
This is not merely saying that both sets are infinite but one somehow "feels" bigger.
There is no possible one-to-one correspondence between the natural numbers and the real numbers.
No matter how cleverly we try to list the real numbers, some will always escape the list.
Cantor had proved something extraordinary:
Infinity has different sizes.
Are There More Real Numbers Than Fractions?
Yes.
And there are vastly more.
This can initially seem extraordinary because rational numbers are dense on the number line.
Between any two different real numbers, we can find a rational number.
Yet despite appearing everywhere, the rational numbers form only a countable infinity.
The irrational numbers make up the overwhelming remainder.
An interesting way of expressing this is:
There are more irrational numbers than rational numbers.
Not merely a few more.
They belong to a genuinely larger infinity.
A Strange Thought Experiment
Imagine choosing a real number completely at random from between 0 and 1.
What is the probability that it is rational?
Mathematically, the answer is:
That does not mean there are no rational numbers.
There are infinitely many.
But compared with the uncountably many real numbers, the rational numbers occupy a set of measure zero.
That is an extraordinary example of why infinity cannot be understood simply by imagining "a very large number".
Infinity Is Not Just One Destination
School mathematics sometimes gives the impression that numbers simply grow:
10
1,000
1,000,000
10^100
and eventually we somehow arrive at infinity.
But infinity does not work like that.
No matter how enormous a finite number becomes, adding 1 produces something larger.
There is never a "largest finite number".
Infinity describes something fundamentally different.
And once we begin studying infinite sets, we discover that even infinity itself does not have a single size.
And Cantor Went Further
Perhaps the most remarkable part is that the real numbers are not the end of the story.
Cantor proved that for any set, we can construct another set that has a larger cardinality.
Take a set S.
Now form its power set, written P(S), consisting of every possible subset of S.
Cantor's theorem tells us:
|P(S)| > |S|
This means that if we start with one infinity, mathematics can construct a larger one.
Then another.
Then another.
There is no largest infinity.
So we do not merely have:
finite numbers -> infinity.
We have an entire hierarchy of infinities.
That is a remarkable place to arrive after beginning with imaginary hotel rooms.
A Few Challenges for Students
This topic lends itself beautifully to investigation rather than simply reading about the result.
Challenge 1: Counting the integers
Find your own rule that pairs:
1, 2, 3, 4, ...
with:
..., -2, -1, 0, 1, 2, ...
Can you produce a formula describing the relationship?
Challenge 2: Count the fractions
Draw a grid of fractions using numerators and denominators from 1 to 10.
Try travelling through it diagonally.
Which fractions appear more than once?
How could you eliminate duplicates?
Can you convince yourself that every positive rational number will eventually be reached?
Challenge 3: Infinity + infinity
Imagine Hilbert's Hotel receives two infinitely long coaches.
Can everyone still be accommodated?
What about 10 infinitely long coaches?
What about an infinite number of infinitely long coaches?
The answers become increasingly interesting.
Challenge 4: Create your own diagonal proof
Write down ten different non-terminating decimals between 0 and 1.
Construct another decimal by changing the first digit of number 1, the second digit of number 2, and so on.
Show why your new number cannot be any of the ten numbers listed.
Then imagine extending the argument infinitely.
Questions Worth Discussing
This subject produces some excellent mathematical conversations.
Is infinity a number?
Not in the ordinary sense in which 17 or 4.5 is a number.
The infinity symbol is used in several different mathematical contexts, while set theory uses cardinal numbers to describe the sizes of infinite sets.
Can infinity + 1 equal infinity?
For countably infinite cardinalities:
yes.
Adding one new member does not change the cardinality.
Are there more even numbers than odd numbers?
No.
Both are countably infinite.
Are there more integers than positive integers?
No.
Again, both are countably infinite.
Are there more fractions than integers?
Surprisingly, no.
Both are countably infinite.
Are there more real numbers than fractions?
Yes.
The real numbers are uncountable and therefore form a strictly larger infinity.
Is there a biggest infinity?
No.
Cantor's theorem provides a way of constructing an even larger cardinality from any set we begin with.
Why Teach Something That Is Not on the Syllabus?
No GCSE examination is likely to ask a student to prove Cantor's theorem.
That is not really the point.
There is enormous educational value in occasionally allowing mathematics to escape the syllabus.
Students spend much of their school mathematics learning how to solve particular types of problem:
factorise this expression;
differentiate this function;
solve this equation;
calculate this probability.
All of those skills matter.
But mathematics is also about ideas.
Cantor's work on infinity shows students mathematics doing something much deeper: questioning what apparently obvious words such as number, size, same, more and infinite actually mean.
It demonstrates that rigorous reasoning sometimes leads us somewhere that intuition never would.
And that is mathematics at its best.
Conclusion: Infinity Is Only the Beginning
Hilbert's Hotel begins as an entertaining paradox.
A full hotel accepts another guest.
Then infinitely many guests.
Then we discover that the even numbers are somehow the same size as all the counting numbers.
Then that the fractions are also the same size.
Just as we begin to think that perhaps every infinity behaves this way, Cantor pulls the mathematical rug from beneath us.
The real numbers cannot be counted.
Their infinity is larger.
And even that infinity is not the largest possible infinity.
For students, this is a wonderful example of what lies beyond the examination specification.
You do not need pages of algebra or years of university mathematics to appreciate the central idea.
You simply need to be willing to ask an apparently innocent question:
How big is infinity?
The surprising answer is:
Which infinity do you mean?
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