Exactly true, @confusedpublic. Wheeler's insight didn't pan out as literally true, but it helped Feynman polish off QED and the basics of quantum field theory. And the point is that all "electron particles" are actually connected -- they're all excitations of the same field -- which explains why they share these properties.
The observational evidence very much is there to say that there's only one electron/positron, albeit not quite in the form Wheeler came up with. And it's not an electron in the naïve way basic quantum mechanics thinks of it.
Instead there's only one electron/positron quantum field. Of course all particle-like excitations of the field have (certain) identical properties, like spin, rest-mass, and so on; they're all really properties of the field.
It's not really all that much nonsense once you make the jump to quantum field theory. A lot of "quantum weirdness" looks a lot less weird once you're thinking in terms of relativistic quantum fields.
Oh, but that would mean giving up a lot of hand-waving and woo-woo and having to do some math. Yeah, can't have that.
Let's try the Poincaré map again:
Differential equations are HARD, so instead of trying to solve a big system of differential equations Poincaré had a great idea. To simplify even further, I'll talk in terms of a three-body system composed of the Sun, the Earth, and a small asteroid.
The asteroid is so small it's not really going to affect the Earth or Sun that much, so we can just say that the Earth goes around the Sun, making a circle (almost) in a plane. In fact, we can even pick our coordinates to rotate with the Earth, so that the Sun is at (0,0,0) and the Earth is at (1,0,0) forever. Now we're interested in the path our asteroid takes.
But like I said above: differential equations are HARD! So instead of really working out the path, Poincaré chose to study where the asteroid crosses from above the Earth-Sun plane tobelow it. If you know a point where the asteroid crosses then it's actually not too hard to figure out where it's going to cross from above to below again.
This is the Poincaré map: a function from the plane to itself saying that if you cross at point p you're next going to cross at point f(p), and then f(f(p)), and so on. Studying how iterations of a function behave is a lot easier than solving differential equations, and can tell you a lot of the same qualitative information.
If anyone is interested on exactly why this works, at least in the case of n-dimensional balls (spaces like filled circles, filled spheres, and higher-dimensional analogues), here goes:
Let's say that you've got a continuous function f from a ball back into itself with no fixed point (this is what I'm saying actually doesn't exist). Then for every point p in the ball you can draw a ray from p through its image f(p) and off into the distance. This ray is parallel to some unique ray from the origin to a point on the surface of the (n-1)-dimensional sphere; points in 2-D balls like maps give us points on the 1-D sphere, or "circle". The assumption that f is continuous tells us that you've now got a continuous function from your ball to the (n-1)-dimensional sphere.
Here's the Hard Bit: because you've sent the ball back into itself, the image of this function covers the whole sphere. I won't prove this here.
But the problem is this: the ball doesn't have a hole, but the sphere does; it's hollow! And we know that there's no continuous function from a space without a hole onto all of a space with a hole. You'd have to punch a hole in your first space to make it work, and continuity says that's not allowed.
Great nitpick! (this is actually a compliment in math)
Right; I said "starting with", and I covered the rest of the argument where it came up earlier in the comments tree.
Don't follow the rows one-by-one; follow the path that snakes along the diagonal.
A better one is Everything and More, by David Foster Wallace.
nitpick: a property of cardinal numbers is that they can be associated with sets of things.
No; the same argument would work for rational numbers, and there are just as many whole numbers as there are rationals.
Bullshit, starting with the idea that fractions are irrational numbers, which is the exact opposite of true.
As David Foster Wallace wrote in Everything and More, suggesting that thinking about infinity drove Cantor insane is like saying St. George was killed by a dragon; it's not only wrong, it's insulting.
It doesn't apply at all to physical balls. In fact, the pieces are such bizarre shapes that it's basically impossible to actually describe them, but you can prove that they exist by using the Axiom of Choice.
Not any less dense; the resulting balls are identical to the original ball.
The point is that some of the axioms we conventionally use lead to really weird effects in some cases. Unfortunately if you throw out the thing that leads to B-T you lose the Axiom of Choice, which says that if you've got a bunch of sets you can pick one element from each one.
Since I'm decidedly pro-Choice, I'll accept the Banach-Tarski "paradox" and its multiplying oranges. It takes a lot of balls to side with Choice.
Oh, and yeah, math students are pretty much all like that.
I also liked that they at least tried to get some science in the fiction. Yes, you're absolutely right that a Faraday cage wouldn't work like that (I was thinking the exact same thing), but the fact that they even said the words "Faraday cage" puts it above a lot of sci-fi.
"Best worst" is a great way to put it. It was corny as hell, but it had a heart to it Mission: Impossible just didn't, and it looked pretty cool, too. I had fun, and that's what counts.
The other reason there are many proofs is that there are actually many versions of the theorem, which apply to subtly different cases.
Now away from my office I can go to YouTube: Outside In actually explains more about what's going on.
