Does this statement represent a set? Does the following statement represent a set?
"The collection of all the stars in the universe?". I think it represents a set. Please explain if someone agrees or disagrees.
 A: Whether or not it's a set really depends on the context in which you are asking the question.
The answer is "no" if you're doing mathematics, since there's no mathematical definition of a star (in the sky).
But questions like this frequently appear at the beginning of a class where you are first learning informally about sets in mathematics. A set might be described as "a collection of things". The answer would be "yes",  as long as you're willing to agree that everyone knows what a star is (something astronomers might argue about). Then the informal collection of all of them is a set. The sun is a member, so is alpha centauri. The moon is not. Your computer is not.
Of course defining a set as a "collection of things" is OK for  informal understanding but won't do as a mathematical statement unless you had previously defined "collection" ...
A: The question as it stands lacks mathematical rigour; the statement that it's a set is one of those things you could justifiably say is not true, nor even false. To make it something for which there's an answer, we need to remember that a set theory is a specific set of axioms in a specific language. Once you fix your language and theory, whether it's a set comes down to whether "$\{x:x\textrm{ is a star}\}$ exists" is a theorem of that theory. ZF, alone, cannot even phrase the predicate in question, unless you come up with a way to define "$x\textrm{ is a star}$" only in terms of $\in$ and $=$. Even if you did, there are many models of set theory, but only the one universe of stars, and the question of which models "actually" included the stars would remain essentially metatheoretic.
One might grant that it's a philosophically interesting question as to whether some sets are identical with physical objects; or even whether the urelements of a similar theory are. But this is a philosophical question, not a mathematical one.
A: To define a set, you just need to define its elements.
here,  elements are stars or suns.
the elements are well-defined, so we can speak about the set $S $ of these elements.
for example, $($vega, the  sun$) \in S^2 $.
