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

There’s a rule that set theorists use when talking about sets with infinitely many objects in them. They say that such sets are the same size if there exists some way to pair up members of one set with members of the other set so that neither set has any members left out. For every item in set A you could ever name, you should be able to name one, and only one, counterpart in set B, and vice versa.

So one weird result of this rule, is that the set of positive even numbers, has the same size as the set of positive whole numbers (aka the natural numbers, or N.) You might initially guess that one is twice as big as the other. But you can easily come up with a rule between these sets which matches their members up so none is left out. For every member of the whole numbers, multiply it by 2 to find its partner in the even numbers. For every member of the even numbers, divide by 2 to find its partner in the whole numbers.

When you can devise a rule like this which perfectly assigns partners to partners between two sets, we call this a “one to one ~~mapping~~correspondence”, or a “bijection”.

Now there’s a tricky question to ask: Between any 2 infinite sets, is there always a bijection?

A mathematician named Georg Cantor proved that sometimes there isn’t, using his now-famous “diagonal argument.” Specifically, he proved that there’s no bijection between the natural numbers N and the real numbers R. No matter what scheme you might propose to assign every whole whole number a partner in the real numbers, you can always come up with more real numbers which are unpartnered. This means that the size of R is strictly greater than the size of N.