Last week I found something that blew my mind. It's not new to the mathematical world - it's just new to me. I don't have anyone here to share it with (no friends or family are interested enough to sit through this), so I'm sharing it here.
Background:
The set of all integers is "countably infinite". This means it's infinite, but there is a counting order - from wherever you are in the list, you know exactly which number is next.
The set of all real numbers is "uncountably infinite" - it's infinitely larger than "countably infinite" because there's no next number. You can pick any two numbers and no matter how close they are, you can always find more numbers in between. If I say 1.4 follows 1.3, I can see that 1.35 falls between the two, and 1.31 comes before 1.35, but 1.3000000000...<insert as many zeros as you want>...1 comes before that.
What I discovered:
Any finitely-bounded subset of the real number set is just as large as the real number set itself:
You can pick any interval in the real number line - pick a low number, L, and a high number, H. There are uncountably infinite numbers between L and H. You can prove this by defining a bijection:
For any X between L and H, we can map it to a number, R, on the Real number line with:
R = Tan(-pi/2 + pi*(X - L)/(H-L))
We can convert back to X from R with:
X = (Atan(R)+pi/2)*(H-L)/pi+L
This associates every real number from -infinity to infinity to a real number between H and L. It means that the size of the set of real numbers between any L and H that you choose is just as large as the size of the set of all real numbers from -infinity to infinity. There are uncountably infinite real numbers from -infinity to infinity. There are also uncountably infinite numbers between 0 and 1, and between 1.000000000001 and 1.000000000002, etc.
This also means that you could say:
There are uncountably infinite real numbers between 0 and 1.
There are uncountably infinite real numbers between 1 and 2.
There are uncountably infinite real numbers between 2 and 3.
....
So the set of real numbers can be divided into a countably infinite number of subsets of uncountably infinite numbers.
The difference between countable and uncountable infinities:
I thought that one difference was that when you pick two numbers, say 1 and 10, the set of all natural numbers has a finite number of elements in that range (10 numbers). But the set of all real numbers has an uncountably infinite number of elements in that range. I thought, "Aha. This is a major difference between the two". But then I discovered:
The set of all fractions is just as large as the set of all natural numbers.
Cantor proved this by defining an order for counting all fractions. Here's an illustration: Link.
Any finitely-bounded subset of the set of fractions is just as large as the set of all fractions:
Given the fractions a/b < c/d, we can pick any fraction 0 ≤ x/y ≤ 1 and use it to find a fraction in [a/b, c/d] with:
new fraction = a/b + (x/y)*(c/d - a/b)
Since there are countably infinite ways to choose x and y, this means that you can pick any two elements in the set of all fractions and there will be a countably infinite number of fractions between the two.
What blew my mind:
Suddenly, we have taken a countably infinite set, the set of natural numbers, which were well-ordered and could be divided into intervals with finite elements (e.g. elements 1 to 10) and turned it into a set which is still countably infinite but which has a countably infinite number of elements in any interval you choose!
Now, just like we did with the set of real numbers, we can say: the set of real numbers can be divided into a countably infinite number of subsets of countably infinite numbers.
A recap:
The set of real numbers uncountably infinite in size. Every subset defined by a lower and upper element is also uncountably infinite in size.
The set of natural numbers is countably infinite in size. Every subset defined by a lower and upper element is finite in size.
The set of fractions is countably infinite in size. Every subset defined by a lower and upper element is countably infinite in size.
And yet, the set of fractions is the same size as the set of natural numbers.
I just think this is so cool.
Now, do I say "Thank you for letting me infodump" or "Thank you for listening to my Ted Talk"?
Background:
The set of all integers is "countably infinite". This means it's infinite, but there is a counting order - from wherever you are in the list, you know exactly which number is next.
The set of all real numbers is "uncountably infinite" - it's infinitely larger than "countably infinite" because there's no next number. You can pick any two numbers and no matter how close they are, you can always find more numbers in between. If I say 1.4 follows 1.3, I can see that 1.35 falls between the two, and 1.31 comes before 1.35, but 1.3000000000...<insert as many zeros as you want>...1 comes before that.
What I discovered:
Any finitely-bounded subset of the real number set is just as large as the real number set itself:
You can pick any interval in the real number line - pick a low number, L, and a high number, H. There are uncountably infinite numbers between L and H. You can prove this by defining a bijection:
For any X between L and H, we can map it to a number, R, on the Real number line with:
R = Tan(-pi/2 + pi*(X - L)/(H-L))
We can convert back to X from R with:
X = (Atan(R)+pi/2)*(H-L)/pi+L
This associates every real number from -infinity to infinity to a real number between H and L. It means that the size of the set of real numbers between any L and H that you choose is just as large as the size of the set of all real numbers from -infinity to infinity. There are uncountably infinite real numbers from -infinity to infinity. There are also uncountably infinite numbers between 0 and 1, and between 1.000000000001 and 1.000000000002, etc.
This also means that you could say:
There are uncountably infinite real numbers between 0 and 1.
There are uncountably infinite real numbers between 1 and 2.
There are uncountably infinite real numbers between 2 and 3.
....
So the set of real numbers can be divided into a countably infinite number of subsets of uncountably infinite numbers.
The difference between countable and uncountable infinities:
I thought that one difference was that when you pick two numbers, say 1 and 10, the set of all natural numbers has a finite number of elements in that range (10 numbers). But the set of all real numbers has an uncountably infinite number of elements in that range. I thought, "Aha. This is a major difference between the two". But then I discovered:
The set of all fractions is just as large as the set of all natural numbers.
Cantor proved this by defining an order for counting all fractions. Here's an illustration: Link.
Any finitely-bounded subset of the set of fractions is just as large as the set of all fractions:
Given the fractions a/b < c/d, we can pick any fraction 0 ≤ x/y ≤ 1 and use it to find a fraction in [a/b, c/d] with:
new fraction = a/b + (x/y)*(c/d - a/b)
Since there are countably infinite ways to choose x and y, this means that you can pick any two elements in the set of all fractions and there will be a countably infinite number of fractions between the two.
What blew my mind:
Suddenly, we have taken a countably infinite set, the set of natural numbers, which were well-ordered and could be divided into intervals with finite elements (e.g. elements 1 to 10) and turned it into a set which is still countably infinite but which has a countably infinite number of elements in any interval you choose!
Now, just like we did with the set of real numbers, we can say: the set of real numbers can be divided into a countably infinite number of subsets of countably infinite numbers.
A recap:
The set of real numbers uncountably infinite in size. Every subset defined by a lower and upper element is also uncountably infinite in size.
The set of natural numbers is countably infinite in size. Every subset defined by a lower and upper element is finite in size.
The set of fractions is countably infinite in size. Every subset defined by a lower and upper element is countably infinite in size.
And yet, the set of fractions is the same size as the set of natural numbers.
I just think this is so cool.
Now, do I say "Thank you for letting me infodump" or "Thank you for listening to my Ted Talk"?
