By the end of the 19th century, the mathematicians had called “set” a collection of any objects. For example, the set of the natural numbers, the set of integers, the set of even numbers, the set of the real numbers, the set of any sets, the set of all the sets. We can give further similar examples. All the mathematicians had no doubt about that the unique condition to be a set was to gather any objects by the time Bertrand Russell’s paradox emerged. Russell had proved that when the term "set" is defined as “a collection of any objects”, there emerges a paradox in the set theory . Now, let’s examine Russell’s paradox and its proof: Assume that a collection of any objects is a set. In that case, the collection of all the sets is a set. We denote this set by $X$. Hence, any set is an element of the set $X$ i.e., if $A$ is a set, then $A\in{X}$. Since $X$ is also a set, then $X\in{X}$. Let’s construct a subset of $X$:
$Y=\{A\in{X}\:|\:A\notin{A}\}$.
Which proposition of the two is the true one $Y\in{Y}$ or $Y\notin{Y}$?
i) Let’s assume that the proposition $Y\in{Y}$ is the true one. In that case, since any element of $Y$ is a set that is not an element of itself, the proposition $Y\notin{Y}$ is true.
ii) Let’s assume that the proposition $Y\notin{Y}$ is the true one. In that case, according to the definition of $Y$, the proposition $Y\in{Y}$ is true. As a result, the following proposition has been proved: