It's a little confusing though, at least for me. There are infinite integers, and "doubly-infinite" reals. But there are also only "doubly-infinite" complex num

It's a little confusing though, at least for me. There are infinite integers, and "doubly-infinite" reals. But there are also only "doubly-infinite" complex numbers. IMO the easiest way to illustrate different, and infinite number of, infinities is to think of sets. Suppose you have N-infinite number of things, say triple-infinite to use the grandparent's lingo. Now the number of sets that you can create from these triple-infinite objects is greater than triple-infinite, and it's called quad-infinite. You can prove that indirectly. Assume that there is a one to one mapping of the original object to the sets. That means that some of the objects are mapped to sets that the objects are members of (let's call these red objects), others not (let's call them blue objects). Does the set of blue objects get paired with a blue or a red object ? Cannot be either, so we've reached a contradiction, ergo there are always more subsets of things than things.