Arbitrary union definition in set theory I am reading Enderton's "Elements of Set Theory". He defines the union operation as
$$\cup A = \{ x \;|\; x \text{ belongs to some member of } A\} = \{x \;|\; (\exists b \in A) x \in b\}$$
Maybe I am missing a subtle point from earlier on in the text, but what if the set in question is not a "set of sets"? For example, if $A = \{1,2,3\}$, then is $\cup A = \emptyset$?
 A: When you're working in a system built out of the set theory, everything is a set. I assume the book will cover this later on, but one way of "building natural numbers" out of sets is called Von Neumann ordinals and the construction is as follows:
Let $0 \equiv \emptyset$. Let the number $n \equiv n - 1 \cup \{ n - 1\}$. That is, under this system:
\begin{align*}
&1 \equiv 0 \cup \{ 0 \} = \emptyset \cup \{\emptyset\} = \{\emptyset\} \\
&2 \equiv 1 \cup \{1 \} = \{\emptyset\} \cup \{\{\emptyset\}\} = \{\emptyset, \{\emptyset\}\} \\
&3 \equiv 2 \cup \{2\} =  \dots = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}
\end{align*}
The point of the representation is that 0 is a set with zero elements, 1 is a set with 1 element, 2 is a set with 2 elements and so on. So we can define the "size of a set" as "the natural number it is in bijection with".
This goes down a rabbit hole, and we can then ask "well, now that I have the natural numbers, how do I define addition? Multiplication? What about the integers? rationals? reals?"
We can construct all of these things, and a set theory book usually describes these encodings in one of the later chapters.
But the point is that at this level, all the "stuff of math" is sets, and you can always write expressions such as $1 \ \cup 2 \cup 3 = \{\emptyset\} \cup 
\{ \emptyset, \{\emptyset\}\} \cup
 \{ \emptyset, \{\emptyset\},\{ \emptyset, \{\emptyset\}\} \} = \{ \emptyset, \{\emptyset\},\{ \emptyset, \{\emptyset\}\}\} = 3 $!
A: As Lord Shark the Unknown indicates, in nearly all systems of set theory, every object under the sun is a set.  In particular, the elements of any set are, in turn, sets themselves.  For the specific example you give, i.e.
$$ \cup \{ 1,2,3 \}, $$
we need to be able to describe the natural numbers as sets.  On page 67 of Enderton's text, the construction is outlined.  He defines
\begin{align}
0 &= \varnothing \\
1 &= \{0\} = \{ \varnothing \} \\
2 &= \{0,1\} = \{ \varnothing, \{\varnothing\} \} \\
3 &= \{0,1,2\} = \{ \varnothing, \{\varnothing\}, \{ \varnothing, \{\varnothing\} \}\},
\end{align}
and so on.  The "and so on" is explained in somewhat more detail in chapter 4 of the text (starting on page 66).  Note that we could write $2 = \{\varnothing, \{\varnothing\} \}$ over and over again, but it is likely easier to see what is going on if we work one level of abstraction higher.  That is, $2 = \{0,1\}$.  Once we accept this abstraction, we have
$$
\cup\{1,2,3\}
= 1 \cup 2 \cup 3
= \{0\} \cup \{0,1\} \cup \{0,1,2\}
= \{0,1,2\}
= 3.
$$
