Sizes of infinities

Question

How is it that some infinities are bigger than other infinities and also If I have an infinite amount of apples and an infinite amount of planet earths then which will have the greater mass? (My intuition is leaning more towards obviously planet earths but why is that, should they both not be the same?

Answer 1

Imagine that you have an infinite amount of boxes all lined up next to each other, starting from outside of your house and continuing forever. Each box has room for exactly one number.

Now take the numbers $1,2,3,4,5,\ldots$

There are infinitely many of these numbers, but you can take the first of them and put it in the first box, the next number you put in the second box, and so on. There's a box for every number.

Similarly, if I gave you the numbers $\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots$, you would still be able to put them into your infinite number of boxes. One way is to first take $0$ and put it in the first box, then take $1$ and put it in the second box, $-1$ in the third box and so forth, following the pattern $$0 \rightarrow \text{box } 1$$ $$1 \rightarrow \text{box } 2$$ $$-1 \rightarrow \text{box } 3$$ $$2 \rightarrow \text{box } 4$$ $$-2 \rightarrow \text{box } 5$$ $$3 \rightarrow \text{box } 6$$ $$-3 \rightarrow \text{box } 7$$ and so on.

So it's like that both the set of numbers $1,2,3,4,\ldots$ and the set of numbers $\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots$ fit into the same infinite chain of boxes, which is why it makes sense to think of these two sets of numbers as representing the same infinity! We say that the two sets have the same size.

Maybe you're thinking that certainly the numbers $1,2,3,\ldots$ are only a part of the numbers $\ldots,-4,-3,-2,-1,0,1,2,3,\ldots$, and therefore they must represent a smaller infinity. But imagine that you had two infinite chains of boxes perfectly aligned with each other, instead of having just one. Maybe all boxes in the first chain are red, and the others are blue.

I'm putting all the numbers $1,2,3,\ldots$ into either the red or the blue boxes, but I'm not telling you which ones, and I put all the numbers $\ldots,-3,-2,-1,0,1,2,3,\ldots$ into the colored boxes that I didn't use. Now I have two infinite chains of boxes, aligned perfectly next to each other. You can match each red box with exactly one blue box, and vice versa. You don't know which numbers are in which infinite chain of boxes, and it surely looks like that there are equally many boxes in both chains. This is why it makes sense to speak of the two different sets of numbers as being "equally infinite". You cannot tell the two infinities apart, when you line up the sets of numbers next to each other.

The first natural thought is then: You can put any infinite set of numbers in boxes like this, and per the same argument, any infinite set of numbers will have the same size as $1,2,3,\ldots$.

But this is not true. You can't put any infinite set of numbers into the boxes like this. For an example of a set, where you cannot do this, think of the interval $[0,1]$. This is all numbers between $0$ and $1$. You can prove mathematically that no matter how you try, you can't place all numbers in this interval into the boxes. Feel free to try. What number would you put into the first box? And what number will you then put into the second box? And so on.

If you don't have a lot of experience with these concepts, this should sound weird to you. But never the less, one says that since the numbers in the interval $[0,1]$ cannot all be placed in a box as before (there will always be numbers left, if you try), this set of numbers represents a bigger infinity.

This is an answer to your first question that some infinities are bigger than others.