Final answer:
The question pertains to Russell's Paradox, which led to fundamental developments in set theory due to the contradiction it presents in naive set theory.
Step-by-step explanation:
The question "Consider the set of all sets which are not members of themselves; is this set a member of itself or not?" is known as Russell's Paradox. This paradox arises within naive set theory by considering the set of all sets that do not contain themselves. If such a set were to exist, it would both be a member and not a member of itself, leading to a contradiction. This is a crucial problem in the foundation of mathematics, directly prompting significant advancements in set theory to address such paradoxes.