Quote:

Originally Posted by

**Kodoku**
You have to be careful when talking about unqualified 'size', though - while the naturals and rationals have the same cardinal number and thus have the same size in that sense, there is also a sense in which one is clearly bigger than the other. The naturals are a strict subset of the rationals, which is a reasonable sense in which they're strictly smaller in size.

Agreed that I should've used (or at least mentioned) cardinality, but that doesn't mean one is bigger than the other.

Think of it this way:

We have two infinite sets: A and B. Let us imagine a process (isn't that word loaded after reading this thread...) wherein B systematically takes its elements and brags about them to A. Once it uses an element, it will never show it again.

If A, in turn, can counter anything B shows with an element of its own that it has never used before, then:

|A| >= |B|

And, if we reverse this process, with A showing its elements to B, and B counteracting, then we have:

|B| >= |A|

Well, if both of those things are true, the only alternative is:

|A| == |B|

**THIS IS NOT A PROOF.** This is a way of thinking.

The problem people have is this thought:

"If every element in A is also in B, but B has elements beyond that, then B

*must* be bigger."

This is a

*symptom* of finite sets, not an actual rule. We equate them in our heads because there's a 1-to-1 relationship in the finite space, but they are not the same.