Is it really legitimate to denote non-existence of an element as that element being an element of the empty set? My question is simple, but it bugs me...
I've seen many people using e.g. $a\in\emptyset$ to state that element $a$ does not exist... Is this legitimate? And why? I thought nothing can be an element of the empty set, by definition.
 A: I wouldn't recommend this as a general notation for "$a$ does not exist" (or more precisely "there is no $a$ with the properties we have assumed"). Mostly because it is somewhat confusing, but also because the notation leaves it implicit which properties it is we have concluded $a$ cannot have.
Formally, $a\in\varnothing$ is simply another way to say "false" or "contradiction", because $\varnothing$ is defined such that $x\in\varnothing$ is always false.
This means, that if as part of an argument we have assumed that $a$ with such-and-such properties exists, and after some reasoning we reach $a\in\varnothing$, we can conclude that our assumption must have been false -- we have proved by contradiction that no $a$ can have these properties.
However, if we have assumed that $a$ with some properties exist, and that $b$ with some other properties exist, then if we reach $a\in\varnothing$, that doesn't tell us that it was $a$ in particular that doesn't exist -- it might as well have been the assumed properties on $b$ that lead to this contradiction.
A: Consider the following situation. The set $A=\{x\in \mathbb R \mid x^2 <0\}$ is the empty set. Now, suppose you want to find the real solutions of $x^2+1=0$. So, you can argue as follows: Assume that $y$ is a solution. Then $y^2=-1<0$ and therefore $y\in A$. But $A=\emptyset$ and so $y\notin A$, a contradiction. The contradiction stems from the assumption that a solution $y$ to $x^2+1=0$ exists and so we may conclude that such a solution does not exist. 
