The Capgras Delusion is a psychological disorder where you think everybody you know has been replaced with an impostor.
A cool hypothesis to explain this (and one which has some evidence, I think) is that the neural pathways from the part of the brain which recognises people to the part of the brain which causes an emotional response has been in some way severed. So you see your loved ones and yet don't feel any kind of emotions towards them. Your brain rationalises this by insisting that they're somebody different.
The Fregoli delusion is the opposite. You think that different people are in fact one person. It's pretty funny, really.
Originally Posted by Marvo
There are different levels of infinity. In this case, you have infinity times infinity decimals.
Ah well, technically the infinity of the whole numbers (let's call it N0) multiplied by itself should be the number of pairs of whole numbers (in the same way that multiplying 2 by 3 means count how many points there are in a 2 by 3 grid), in other words the number of whole points in the first quadrant of the plane, but this turns out just to be equal to the original infinity; i.e. N0*N0 = N0. That's surprising fact #2 about infinity.
The infinity of decimals, call it N1, turns out that it should equal, if anything, 2 to the power of N0. But this turn out to be impossible to prove from the axioms of number theory or set theory, and in fact you can consistently take it to be either true or false. Extremely surprising fact about infinity #3.
Originally Posted by Dianeva
There are still an infinite number of decimal numbers left after the mapping.... so there must be infinitely more decimal numbers than positive integers.
Technically this reason doesn't work. For example, you could double every whole number. This mapping sends every whole number to every even number inside the whole numbers. There are an infinity quantity of whole numbers still left after the mapping, but you can't then conclude that there are infinitely more whole numbers than whole numbers.
In the same way you could map the infinite quantity of decimal numbers into the space between, say, 1 and 1.
The problem is that you're trying to show that there are some decimals left untouched no matter what mapping you use, not that any specific map fails (it's always possible to come up with maps which fail). In other words, given a map from the whole numbers to the decimals, you need to find a methodical way of producing a decimal which can't be touched anywhere on the list. There's an ingenious way of doing this.
That's the reasoning anyway. And in a way this makes sense, perhaps it's useful in some situations. But I think, on a more commonsense level, one infinity can't be greater than another. Infinity is a concept, not an actual number, though it can be treated like a number in some circumstances. And the concept of infinity refers to some abstract quantity which is by definition greater than any amount we can possibly name. To say that ∞1 > ∞2 contradicts the definition of infinity. Because by definition nothing can be greater than ∞2.
Perhaps I've become numb through overexposure, but I don't feel there's anything wrong with it. The maths of different infinities does behave in a nice consistent way, anyway. I'd turn what you said on its head and interpret it in the opposite way  these results simply tell us that our intuitions about infinity are inconsistent.
After all... the infinity of the decimals is bigger than the infinity of the whole numbers. But surely we don't want to say the infinity of the whole numbers is not infinite? There's always a bigger whole number than any whole number you can think of. They go on forever and ever. We've gotta call them infinite.


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