Definition of a set

Solution 1:

The intuitive notion of a set is of a collection whose identity is entirely determined by its members. Russell's paradox resulted, in effect, from taking any condition at all to determine a collection of members and thus a set. (Russell formulated his paradox against Gottlob Frege's Basic Law V which has, as a consequence, that every condition determines a set.)

In a way, there is no "one notion of a set." In response to Russell's Paradox (and others, such as the Burali-Forti Paradox), various axiomatizations of set theory have been developed to try to capture and render precise the notion of a set.

Results due to Goedel and Cohen show that widely accepted principles governing sets as captured by the Zermelo–Fraenkel axiomatization fail to decide some interesting and contentious "higher" claims about sets (the Axiom of Choice and the Continuum Hypothesis).

So, in a real way, there is not consensus in the mathematical community about what exactly a set is nor about what principles give a full description of sets. (In practice, most mathematicians use Zermelo–Fraenkel set theory and make appeal to the Axiom of Choice, though many prefer to avoid the later when possible. Some refuse to accept it outright.)

Solution 2:

In very naive set theory (say in the late 19th century), a set was taken to be an arbitrary collection of objects. The difficulty is in telling which things that seem like they should be collections actually are well-defined collections. The paradoxes show that it's unclear whether this concept of set is coherent, although it is the natural-language meaning of the word "set".

Because the concept seems poorly defined, almost all contemporary set theory deals with a more restrictive notion: pure, well-founded sets. These are the sets that can be constructed starting with the empty set and taking powersets and subsets. The only elements of these sets are other sets.

These sets are defined in stages. At the first stage, you only have the empty set. At every larger stage, you add the powerset of every set that has already been constructed. There is one stage for every ordinal number, and the collection of all sets available after stage $\alpha$ is named $V_\alpha$. Symbolically, we have $V_0 = \emptyset$ and, in general, $$ V_\alpha = \bigcup_{\beta < \alpha} P(V_\beta) $$

For each ordinal $\alpha$, $V_\alpha$ is a set. The union $V = \bigcup_\alpha V_\alpha$ is a proper class. The sequence $( V_\alpha )$, indexed by ordinals, is known as the cumulative hierarchy.

The "sets" that mathematicians study and that are formalized in Zermelo-Fraenkel set theory are exactly the sets in $V$. Moreover, the formalization of mathematics into set theory does not require any other sets than those in $V$ (which is why essentially all mathematics can be formalized into ZFC).

So, for all practical mathematical purposes (outside of set theory), the answer to "what is a set" is "a set is an element of $V$".