Every set has a particular property? It is possible to transform any set in the synthetic form?
For example, the set $A=\{0,1,2,3\}   $ can be written as $A=\{x \in \mathbb{N}\,\,|\,\,x<4\}     $
But my question not just limits to numbers, i mean any set.
 A: As clarified in the comments, the question is asking whether every set $A$ can be described in a way that is simpler than the trivial way of listing all the elements of $A$.
The answer to (my interpretation of this slightly vague) question is no.
Regarding finite sets, what the question is asking for would be a "guaranteed" or "perfect" compression algorithm, which is impossible.    Roughly speaking, for any fixed notion of "description" for finite sets of natural numbers (or equivalently for binary strings) there there are not enough simple descriptions available to describe all of the relevant sets.
Regarding infinite sets, the question could be construed as asking for finitary descriptions for all sets of natural numbers.  This is impossible for a similar reason; namely, that for a fixed notion of "finitary description" there are only countably many descriptions available and so Cantor's diagonalization argument can be used to find a set of natural numbers admitting no such description. (The reader should beware that this part of the answer is subtly wrong for a certain formalization of the question, but this caveat does not seem relevant to the original question.)
