In math two sets(think of them as baskets of apples and oranges) are considered to "contain equal amount of elements", if one can pair the elements so that each of the pairs has one side of it from one set and the other side from the other set and after assembling as many pairs as one can, there's no elements left unpaired in either of the sets.
That is to say: one should take a third, empty, basket, and start taping apples and oranges together by placing the pairs to the third, initially empty, basket and one can conclude that there was equal amount of apples and oranges if after running out of apples there are no oranges left.
(Please do take a look at the formal definition, because the examples above were oversimplified. The formal idea is that one defines a function from one set to another and that the function is required to have certain properties.)
What regards the "rationality" part in this post, then that's just for warm-up. The idea is that unlike in the aforementioned example, there's an infinite number of rational numbers, but one can still say that there's exactly as many rational numbers as there are natural numbers. The proof is actually suitable to teenage school children and can be read from here. Just please be sure to understand the proof before reading this posting onwards.
OK, so, now comes the "reality" part. Please notice the scene from the following video, where a whole motorcycle emerges from some small object that fits into a pocket?
Or, the following armory:
Well, actually, mathematically there's nothing wrong with it.
As of April 2011 I don't remember the name of the theorem, nor have I known or even looked up, leave along understand, the proof of it, but I guess if one understands how to fit an infinite number of real numbers between any 2 real numbers, for example, like
then it should be pretty intuitive that there are as "many" numbers between 0 and a million as there are between 0 and 1. After all, there is at least one bijection from set_A=[0,1] to set_B=[0,1000000] and it looks like:
The inverse of it:
The crazy thing restated: for every number between 0 and 1000000 there exists one number between 0 and 1 in a way that none of the numbers from the set [0,1000000] share a pair-mate from the set [0,1] and all of the numbers in the set [0,1] are "used up". Of course, one might use just [0,0.0042] in stead of the [0,1] and one could still pair all of the real numbers from the set [0,1000000], but, the idea is the same and no, I'm not high, may be just a bit mental, but that's for the others to decide. (I've seen worse: check out, what they do in quantum physics.)
In a modern, TRON, way of saying it: pick a pebble from a pocket, let it rearrange itself a little bit and ride a huge bike. Batteries are included.
There's no controversy whatsoever. :-D
Things like "length" and "volume" are quite artificial "parameters". A bit like points in a computer game, or something along those lines. One might as well just define one's own versions of them, which I omit here, because this post is pretty verbal already and people that are much smarter in math than I am or probably ever will be have already defined plenty of smart alternatives and studied their implications already decades ago, if not centuries ago. (Please don't get me wrong: this post wasn't meant to be very serious or extra educational. That's why it's in the soap opera blog. :-)
I really liked the movies, the Ultraviolet and the Aeon Flux, so here are some clips just for fun: