When I was first introduced to sets the common "description" in place of a formal definition used to be something like a set is a collection of objects that do not have to be related in anyway prior to the formation of the set though the often are in mathematics. And a rule of sets was that an element of a set can be represented once and only once e.g. the following is not a set due to the repeat representation of $2$ $$\{ 1, 2, 3, (4-2) \} $$. Suppose I wanted to create a set which contained the representations (denotation) of the number 2: $$\{2, (\frac{2}{1}), (\frac{4}{2}), 5-3, 2k -2(k-1),...\}$$
In the sense(i.e. not necessarily intended as the philosophical Fregean term sense) of the first example the collection above is not a set but in the sense that it is a collection of representations of $2$ it is.
So it would seem that whether something is a set depends on another property, in addition to being a collection and having each element represented once : that is, context.
What would be a better description(not necessarily a definition) of a set if we want to describe its context? We could use the idea of a universe but that itself is conceived of as a set so the universe cannot provide the context exactly can it? For example, I can see how the universe being $\mathbb N$ or $\mathbb R$ can be of help but the universe of $\mathbb N \cup \ $ (Set of Representations of $\mathbb N$) does not seem like it provides much clearity.