Russell's Paradox

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In the late 19th and early 20th centuries mathematicians were looking to put the foundations of mathematics in a rigorous setting to show that the whole of mathematics was consistent with logic which led to an interesting paradox discovered by the philosopher Bertrand Russell. A paradox is just some statement that leads to a logical contradiction. In this post I will try to give an explanation of this paradox in simple terms that anyone can understand.

The fundamental objects of mathematics are called sets. In this post I will denote the set we are using as S. A set is just a collection of objects which are called the elements of the set. Given any object x and any set S we can ask if x is an element of the set S or not. So we see there are only two possibilities either x is an element of S or x is not an element of S. For example if S = {1, 2, 3} then 1, 2 and 3 are elements of S but 5 is not an element of S.

At the time mathematicians were using what is now called naive or intuitive set theory. For us it is not that important what this is all we need to know is that at the time mathematicians allowed for the creation of any set determined by some property. To see an example of this suppose we want to create the set defined by the property that x is an element of S if and only if x is a nonnegative even number. Then it is clear that S should contain all the even nonnegative whole numbers and nothing else. We can write the set of nonnegative even numbers as S = {0, 2, 4, 6, 8, ...} where the ellipses signify to continue on in the same manner. Thus we can see that for example 4 is an element of this set but 7 is not an element.

The idea that we can create a set based off of any property is what led to Russell's Paradox. Suppose we want to construct the set S which consists of every set which does not contain itself as an element. That is we want S to consist of all the sets x such that x is not an element of x. Let us consider why this gives us a logical contradiction. There are only two cases to consider, either S is an element of S or S is not an element of S.

For the first case if we assume that S is an element of S then by the defining property of the set S we should have that S is not an element of S. Since S can't satisfy both of these conditions at the same time we get a contradiction. For the second case if we assume that S is not an element of S then it must follow from the defining property of S that S is an element of S. So once again we get the same contradiction. We have show that in the only two possible cases we get a contradiction and thus we have a paradox!

To deal with this paradox mathematicians have created new axioms of sets that avoid the possibility of constructing the set S which leads to Russell's Paradox. The details of this so called axiomatic set theory are too complicated to go in to in this post so if you are interested in learning more about the paradox and subject you can start with the following
sources.

https://en.wikipedia.org/wiki/Russell%27s_paradox
https://en.wikipedia.org/wiki/Bertrand_Russell
https://en.wikipedia.org/wiki/Set_theory#Axiomatic_set_theory

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The image of Bertrand Russell and was taken from here