Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2.
If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested in their size, relative to each other.
Intuitively, the length of these parts would approach 0, as we increase their number. But, if it became zero when we have infinitely many segments, then that would be a problem. Because then, by adding up infinitely many parts with length 0, I'd get a 0-length segment back.
So, I'm guessing that the lengths of those parts are not really zero, but some value $\epsilon$, bigger than 0, but smaller than any other real number; the value that comes immediately after zero.
Now that we have this value ϵ, then we could say that the segment AB is composed of infinitely many parts of size ϵ. But, what about CD?
- Would it be composed of twice as many parts of size ϵ?
- Would it still be split into infinitely many parts, but their size being 2ϵ?
While I debated this with a friend, she said that "some infinities are bigger than others", and said that the second segment would be split in twice as many parts as AB, all of size ϵ. To which I replied that if we have infinitely many elements, doubling them doesn't matter, they still are infinitely many.
You could argue that I haven't defined how I split these segments. Let's say that we split them in half, and their halves in half, and so on until infinity. I'm aware that defining the splitting method could make all the difference, but I don't know enough maths to reasonably predict how this would affect the parts. I'm also aware that all I've said above could be seen as complete nonsense: nevertheless, I'm asking it anyway, or I won't be able to sleep tonight without knowing the answer. :P