mathematical representation of absence of a value
The distinction between the empty set 
and the number 
is similar to that between NULL and ZERO. For example, the set of real solutions (or informally "the solution") to 
is 
, but the solution to 
is 
.
In my mind there is no need for a concept like NULL in mathematics if you think of NULL as in NULL-pointers.
NULL in this sense is a technical necessity because you cannot un-define a variable: Once a variable has been assigned a value, a certain bit of memory is reserved for this variable and this memory is marked as re-usable only if the variable goes out of scope (simplified speaking).
You cannot say "The variable with this name doesn't exist anymore." without letting it go out of scope, because that would make language interpretation much more complicated without many benefits. Therefore, to indicate that the value of the variable has no meaning, one uses NULL.
What NULL stands for in the end depends upon the programming language: In some it is a special keyword, but in some it is also just a different name for the integer 
.
You can assign an arbitrary value to NULL in mathematics as mentioned in the other replies (
, 
, etc.) but as mathematics has nothing to do with memory allocation there is really no need for such a thing as NULL.
Videos
How do you define a null set?
What is a null set example?
What is a null set called?
From the Fundamental methods of mathematical economics (4th ed.) by Chiang and Wainwright, page 10:
“The smallest possible subset of S is a set that contains no elements at all. Such a set is called the null set, or empty set, denoted by the symbol Ø or {}.”
“The reasoning for considering the null set as a subset of S is quite interesting: If the null set is not a subset of S (Ø ⊄ S), then Ø must contain at least one element 𝑥 such that 𝑥 ∉ S. But since by definition the null set has no element whatsoever, we cannot say that Ø ⊄ S; hence the null set is a subset of S”
Question:
Why do we define a subset this way, leading to the inclusion of the null set? Could we not (more intuitively) define a subset of S: containing at least one element 𝑥 such that 𝑥 ∈ S AND no one element 𝑥 such that 𝑥 ∉ S?
My intuitive thinking:
If I have an apple, an orange, and a kiwi, I usually don’t also go around thinking that I also have a ‘no fruit’. Feels wrong to claim that ‘no element’ is a good description of my set that definitely contains elements.
Edit: Wow, THANK YOU everyone for such a robust discussion. Lots to think on, lots to turn over in my mind.