Fun quiz: where did the infinitely many candies come from? Story 1:
Let there be a bowl $A$ with countably infinite many of candies indexed by $\mathbb{N}$. Let bowl $B$ be empty. 


*

*After $1/2$ unit of time, we take candy number 1 and 2 from $A$ and put them in $B$. Then we eat candy 1 from bowl $B$.  

*After $1/4$ unit of time, we take candy number 3 and 4 from $A$ and put them in $B$. Then we eat candy 2 from bowl B.  

*After $(1/2)^n$ unit of time, we take candy number  $2n-1$ and $2n$ from $A$ and put them in $B$. Then we eat candy $n$ from bowl $B$.


What happens after 1 unit of time? How many candies are there left in $A$, how many in $B$, how many have you eaten? Answer:

 There are no candies in $A$ left. For any candy corresponding to a given natural number $k$, one can compute the time it was eaten. Similarly there are no candies in $B$. We ate as many candies as the cardinality of $\mathbb{N}$. 

Story 2:
Let there be a bowl $A$ with countably infinite many of candies indexed by $\mathbb{N}$. Let bowl $B$ be empty. 


*

*After $1/2$ unit of time, we take candy number 1 and 2 from $A$ and put them in $B$. Then we eat candy 1 from bowl $B$.  

*After $1/4$ unit of time, we take candy number 3 and 4 from $A$ and put them in $B$. Then we eat candy 3 from bowl B.  

*After $(1/2)^n$ unit of time, we take candy number  $2n-1$ and $2n$ from $A$ and put them in $B$. Then we eat candy $2n-1$ from bowl $B$.


What happens after 1 unit of time? How many candies are there left in $A$, how many in $B$, how many have you eaten? Answer:

 As before, there are no candies in $A$ left. All the candies labelled by even numbers are in $B$, so there are countably many. We ate all the candies labelled by odd numbers, so we ate countably many. 

Question: 
The main question is, in both stories we do essentially the same thing: take two candies from $A$ to $B$, then eat one from $B$. The difference is that in the second case we have infinitely many candies left in $B$, but in the first case it was empty after 1 unit of time. So how did this happen?
Imagine this situation: let $X$ be conducting the eating according to story 1 with his labeling, let $Y$ be watching. But $Y$ secretly has a different labeling scheme in his mind, where the candy number $k$ in $X$'s labeling is candy number $2k-1$ in $Y$'s labeling. Then after unit time, according to $Y$, there should be infinitely many candies in $A$ but according to $X$ it should be empty. So the reality depends on the spectator?
 A: So as you said according to person X, there will be no candies left. Lets take a look, then, at what Y observes. 
From person Y's point of view:
First X moves candy 1 and 3 and from bowl A to B, then eats candy 1. Next, X moves candy 5 and 7 (3 and 4 by X's labeling) to B and eats 3. This continues for infinitely many steps. At this point person Y sees every odd candy eaten. Note that according to Y, the even candies never existed. So Y as well sees that there are no candies left in the bowls.
You may also be interested in these two links:
A strange puzzle having two possible solutions
Ross–Littlewood paradox
A: You have essentially proven, with your stories, that $2\cdot \aleph_0 = \aleph_0$, where $\aleph_0=|\mathbb N|$. There is no sense in which either of your two stories is "impossible".
As for your last question with labeling: you described a scenario that cannot happen in real life, and something weird came out as a result. Why would that be surprising?
But OK, let's try to answer the question. So, what we have is an infinite amount of candy, labeled $1,2,3,4\dots$ by $X$, but they are labeled $1,3,5,7,\dots$ by $Y$. This means that there is no candy labeled $2$ in $Y$'s labeling, and the paradox does not exist.
