So the last person got cloned.

Nope. Trying to apply intuition to cardinality of infinite sets tends to lead to nonsense. Here's something to try to demonstrate the point of this.

Take a hotel with an infinite number of rooms labeled room 1, room 2, and so on. Fill up all the rooms. Now add 1 to each room number, so now we have room 2, room 3, and so on.

Nothing happened to any of the rooms other than the fact that we changed the label.

Now break down a wall and add in another room. (Ignore the possible disturbance this could cause to the guests.) Label it room 1.

Note that this set of rooms should now be equivalent to the set of rooms we started with, as we have one room labeled room 1, one room labeled room 2, and so on. There are the same number of people in the hotel, but one room's empty in one but full in the other.

This should be a convincing enough argument that the number of people in a hotel with an infinite number of rooms with 1 person in them each is the number of people in a hotel with an infinite number of rooms with 1 person in them each, save 1 empty room.

Another thing to note is that, in the hotel case with labels 2, 3, etc., ask everyone to leave. You have a bunch of people leaving. In a hotel with labels 1, 2, 3, etc., ask everyone but the person in room 1 to leave. As someone's leaving for every natural number which is at least 2, the same number of people are leaving every time. I just demonstrated that the same number of people are in both hotels, but after the same number of people leave, one person is left over in one and no people are left in the other.

Again, don't try to take this intuitively, that can lead to problems.

And, a couple pages back:

Also, a very eminent member of this society has given thorough proof that 1+1 = 1

1+1 does not equal 1 because 1=2.

Consider the following proof:

Let a=b where a and b are real numbers,

then ab=b^{2}

so ab-a^{2}=b^{2}-a^{2}.

Factorising both sides gives us

a(b-a)=(b+a)(b-a)

and cancelling out the common factor of b-a leads to

a=b+a.

Choosing a=b=1 we get 1=2.

Hooray for logical errors. (b-a)=0 because b=a. Substituting 4 for a to prove my point... b-4=0 because b=4.

Also, Beorn. The picture you posted breaks down because we define those partially by comparing them to real-world examples (if we didn't initially define words by using physical examples, how could we develop or learn language in the first place?). One object along with one more of that object makes two of that object. Remember, however, that math was initially created with solid objects. Liquids do not act in the same way, so it's pointless to equate solids with liquids (referring to the link you posted).

All of this is pointless though because language (logical equations included) is used to describe reality and communicate ideas. Arguing semantics is an asinine method of debate or point-making. We can communicate what we mean, and **that** is what matters.

1 raindrop + 1 raindrop is still 1 raindrop.

The problem here is that the expression "1+1" is an abstract mathematical concept, as opposed to "the number of apples on a table if we put an apple on the table and then another apple on the table" or "the number of raindrops formed if we bring two raindrops next to each other." (It happens to be equal to the first, and one might need a more rigorous definition of "number of raindrops" to determine if it's equal to the second.)

My point is that parallels to the real world do not constitute a mathematical proof.

EDIT: Beaten!