In naive set theory, any collection of "objects" is a set. Thus, $\{1,2\},\{A,B,C,D\},\mathbf{N}$ are all sets since they are a collection of objects. We say that an object is an element of a set, if it is one of the objects in the set. The word "is" in this sentence refers to "is equal to" as far as I know. Therefore the set of complex numbers $\{ a+ib \ | \ a,b \in \mathbf{R}\}$ also contains the element object $e^{2 \pi i}$. I now wondered if I can visualize any set this way. What I mean by this is, when working with abstract sets, can I visualize this set as a "complete collection" of objects, as in the case of $\mathbf{N}$ or $\mathbf{C}$ of which I don't explicitly know what the elements exactly are? More precisely, can I visualize an abstract set $S$ as a complete list of abstract elements $s$ of $S$? If I could not visualize it this way, then I couldn't imagine what the set $$A:=\{s \in S \ | \phi(s)\}$$ would be for some property $\phi(s)$. Clearly this would be the collection of all $s$ that satisfy this property and are contained in $S$, but it is then impossible to say which elements are in $S$, wouldn't this then be problematic?
Furthermore I wondered about the following: when proving a property for every element of a set $A$, does it then suffice to show it for the elements in one "complete form"? If so, why is that the case? For example, if I wanted to show a property for all elements of {1,2} it would suffice to show it for $1$ and $2$, or if I wanted to show a property for all natural numbers, it would suffice to show it for $1$,$2$,$3$,… or every "$n \in \mathbf{N}$". However, there could be elements that are not "listed" in this set, such as the $e^{2 \pi i}$ example. Why have I shown it for those objects as well? The intuition would be, that they are the same object, and just different "names". Would the formal explanation be that if there is a "hidden" name in the set $S$, then it is equal to one of the "explicit" elements and thus has the exact same properties. Thus, if I have shown a property for all "explicit" elements in $S$, it follows for any form of the elements.
Edit: (Adding a problem) As far as I see it, there are the following two interpretations: $(1)$ A set $S$ contains objects that are denoted by symbols which I will call "initial presentation" of (the objects in) $S$. An example would be writing the natural numbers as $\{1,2,3,4,…\}$ or $\mathbf{C}$ as $\{a+ib \ | \ a,b \in \mathbf{R}\}$. These symbols are the objects of the set and I can rename those objects by using ":=" which defines a new symbol to be the same object as the one on the right side. Thus when writing things like $e^{2 \pi i}:=1$ I define $e^{2 \pi i}$ to be the object $1$. Thus one would have that $e^{2 \pi i}$ is in $\mathbf{C}$ since it is literally just a different name for the same object namely $1$.
$(2)$ A set $S$ again has an initial presentation by the symbols it contains. This time, however, the symbols are not the objects, but rather representing objects of the set. I imagine these objects as points in a space and the symbols standing over the points, where two symbols represent the same object, if they are connectable by a line which would mean that "both symbols refer to the same object". Thus, there would be a point named $1$ and a point named $e^{2 \pi i}$ and they are connected in the set $\mathbf{C}$ since by definition $e^{2 \pi i}=1$. More generally, if I define a new symbol for an existing one, I create a path from the original object, in this case named $1$, to the new symbol, in this case named $e^{2 \pi i}$. This distincts the words "symbol" and "objects" more strictly than $(1)$, but is also harder to understand and also complex to use in my opinion. Now when a set contains one of the symbols, it means that it contains the object represented by this symbol and thus all the other paths as well, which would explain why $e^{2 \pi i} \in \mathbf{C}$. An intuitive problem I see here though is, what is the difference between ${1,2} and any other set containing two elements, say {8,9}? They are both just symbols for two objects (whatever an object is) or more precisely points in my example, that are referred to by different symbols. Thus, in my image, one would have two points with different names in this set and that is the entire difference.
Best Answer
You wrote: "Would the formal explanation be that if there is a 'hidden' name in the set $S$, then it is equal to one of the 'explicit' elements and thus has the exact same properties. Thus, if I have shown a property for all "explicit" elements in $S$, it follows for any form of the elements."
There are no hidden names in $S$, because there are no names in $S$ at all. We may use names to specify what the elements of $S$ are, but the elements of $S$ are the objects named by those names, not the names themselves.
So, for example, if the number 1 is an element of $S$, then $e^{2\pi i}$ is also an element of $S$. That's not because 1 is an "explicit" element and $e^{2\pi i}$ is a "hidden" element that has all the same properties. It's because "1" and "$e^{2\pi i}$" are two different names for the very same object, the number 1. And it is that number, not the name "1", that is an element of $S$.