How are some infinities bigger than others?

Question

This one has got me.

I’m reading a book at the moment that mentioned very briefly that some infinities are bigger than others. The book is unrelated and the author dedicated all of one sentence to the fact – but it’s blown my mind!

I’ve always thought (pretty sure I was always taught) that infinity just is. Something is infinite if it goes on forever, but how can something go on forever more than another?

I’ve tried to google but I’m just not grasping it. How is it that one infinity can be bigger than another?

In: Mathematics

Answer 1

Lets start simple.

Consider a set made up of all the numbers x such that x > 0. Since there are infinite positive numbers, the number of elements in this set is infinite. Lets call this set A.

Now, lets consider a set made up of of all the numbers x such that x >= 0. Since there are infinite positive numbers, the number of elements in this set is infinite. Lets call this set B.

Now, set B will contain all of the numbers in set A, but it will contain 1 number that set A doesn’t: 0. So, the size of set B is 1 + size of set A. Now, since set A is infinite, set B is infinite too, but it is clearly a larger infinite than A.

Lets consider another set, made up of all the numbers x such that x < 0. Since there are infinite negative numbers, the number of elements in this set is infinite. Lets call this set C.

Set C has the same size as set A, since every positive number has a negative equivalent, and thus, it’s a smaller set that B.

If we combine sets C & B together, we get the set of all real numbers, which is more than twice the size of set A, but less than twice the size of set B.

Or, consider a line starting at one point and going out to infinity. If we extend that line backwards from the starting point, we have created a larger infinity. If we choose some new point along that line, and cut away from the starting point to that new point, we have created a smaller infinity. If we draw new lines at different angles from that starting point out to infinity, we have multiplied that infinity. We can do that an infinite number of times, as there is a infinite number of angles in the arc.