BERTRAND RUSSELL'S PARADOX

One of the main approaches to the mathematics is axiomatic tradition. In terms of axiomatic approach, all of the premises of math can be derived from several axioms. When we mean axiom, there are premises that do not require any further premises for its truth. From this view, we will look at one of the most conspicuous paradoxes in formal mathematics.

The paradox is Bertrand Russell Paradox. As it can be understood from its name, the paradox was found out by Bertrand Russell. Consider a set (let’s denote it as S) which includes sets that do not include themselves. So, in set theory we know that belonging and inclusion properties of a set correspond to different things. When we mean inclusion, a set is a subset of itself. This is true for every set. However, belonging is related to the element of a set. A set may contain itself or may not. Thus, a set which includes just 1 and 2 will be appropriate for this big set.

\begin{equation}A = \{1,2\}\end{equation}

When we say a set that includes sets that do not include itself, it represents all the sets that can be created, without restriction, while providing this requirement. For example, set A below is one of the elements of this big set:

As it can be seen set A doesn’t include itself, since the definition of A ({1,2}) is not included in the set of A.

Let’s get the main point. For a moment, assume that the big set S includes itself. Then the paradox will occur after this moment. If the set S includes itself, then according to its definition (S is a set that includes sets that do not include themselves) S becomes a set that doesn’t include itself. Since two states cannot exist at the same time, the Bertrand Russell Paradox occurs. We can represent it in terms mathematical denotation:

\begin{equation}S = \{x:x \notin x\}\end{equation}

This denotation identifies the S set. It basically means that select x such that x does not include in x, thus each set. The main source that leads to this problem is the assumption that there is a universal set which includes all the x values in itself. However, it can be argued that it is impossible to assume the existence of such a universal set. And to restrict this assumption and provide a solution for the paradox, the axiom of selection can be forced in our examination (there are different solutions to the Bertrand Russell Paradox other than axiom of selection as well).

The axiom of selection makes it compulsory to select elements from an established set (let’s say A) when identifying a new set. This restricts the assumption that there exists a set such that includes every element in the universe. We can represent it as:

\begin{equation}S = \{x\in A : x \notin x\}\end{equation}

As an interpretation, this set includes x values that are elements of set A, with the restriction that these x values do not include themselves. So, if we say that what if the set S doesn’t include itself, then we would encounter with two different positions. One possibility is that S is included in S, which leads to the occurrence of Bertrand Russell’s paradox. However, other than this choice, due to the fact that we applied axiom of selection, if we say S is not an element of itself, then S is not an element of set A. Since S includes values of x that are acquired from set of A, and if S, as a set is not included in set A, then this makes sense.