Why can't a set have two elements of the same value? Suppose I have two sets, $A$ and $B$:
$$A = \{1, 2, 3, 4, 5\} \\ B = \{1, 1, 2, 3, 4\}$$
Set $A$ is valid, but set $B$ isn't because not all of its elements are unique. My question is, why can't sets contain duplicate elements?


If we had a 'set' $U$ defined to be $\{1,2,3\}$, could we have another set $V = \{1,1,2,3\}$ where $U\neq V$? 

Yes. We can do this, but using this notation makes it ambiguous and nasty. 
We need to be careful about what we mean when we say $=$. Let's let $=_S$ denote an equivalence relation on sets, and $=_n$ be the usual equivalence relation of numbers. 
It is perfectly consistent to talk about $U,V$, but if we write it down like this we've labeled everything using the weaker relation $=_n$, making it a big mess. I'll show you what I mean:
If $a,b$ are two distinct elements in $V$ where $a=_n 1$ and $b=_n 1$, then $a=_nb$. Furthermore, if we choose the $c \in U$ where $c=_n1$, then $a =_n b =_n c$.
We want  $U\neq _S V$, so we must have that $\{a\} \neq_S \{b\}$. We have no idea whether or not $\{c\}=_S \{a\}$ or $\{b\}$. In fact, it might not even be equal (in the set sense) to either of them.
With this knowlege we can re-label the $1$s in our sets: $U=\{1_c, 2, 3\}$, and $V=\{1_a, 1_b, 2, 3\}$. But we still don't have any idea whether or not $1_c$ as a set element is the same as $1_a$ or $1_b$. The same can be said for $U$'s $2$ and $3$ vs $V$'s $2$ and $3$.
Under this notation we have no idea how $=_S$ works comparing elements across sets.

Here's a better way to do it. Instead of worrying about the set relation, define U and V using an index. 
To start, clarify that every time you write down $a_i$, $i\in \mathbb{N}$, you are talking about the exact same object $a_i$, in every context. This forces the following property: $\{a_i\} =_S \{a_i\}$.
Now define  $a_1 =_n 1\in \mathbb{N}; a_2 =_n 1 \in \mathbb{N}; a_3 =_n 2 \in \mathbb{N}; a_4 =_n 3 \in \mathbb{N}$, and let $U=\{a_1, a_2, a_3, a_4\}, V=\{a_2, a_3, a_4\}$
What we have really done now is make a function from the set $\{1,2,3,4\}\subset \mathbb{N}$ into the set $\{1,2,3\}\subset\mathbb{N}$, and defined $U, V$ in terms of that function. Under this description the $=_S$ relation is well-defined.
