I read a discussion concerning the axiom schema of specification, which I yet take as saying that for every set and a class-defining condition, those elements of the set satisfying this condition extensionally comprise another set, given no antinomies are incurred therein.
First: Does my summary capture the semantic intention of the expression of the axiom in formal logic?
Second: Can you give me some insight into the significance of this statement, and explain why it is granted axiomatic status?
Additional Info:
Is it true that the axiom schema of specification is thought primarily to resolve the antinomies derivable from the Frege-an schema of unrestricted comprehension for classes, and more broadly as a tool with which to confide in the true existence of certain sets smaller than known sets as elements of which the elements of the former exist?