Russell's paradox

Let me tell you a famous story:

There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves.

That is a nice story. But it raises the question: Did the barber shave himself? 

Let's say that he did shave himself. But we see from the story that he shaved only the men in town who did not shave themselves. Therefore, he did not shave himself. But we again see in the story that every man in town either shaved himself or was shaved by the barber. So he did shave himself. We have a contradiction. 

What does that mean?

Maybe it means that the barber lived outside of town. That would be a loophole, except that the story says that he did live in the town, maybe in Seville. Maybe it means that the barber was a woman. Another loophole, except that the story calls the barber "he." So that doesn't work. Maybe there were men who neither shaved themselves nor were shaved by the barber. Nope, the story says, "All of the men in this town either shaved themselves or were shaved by the barber." Maybe there were men who shaved themselves AND were shaved by the barber. After all, "either ... or" is a little ambiguous. But the story goes on to say, "The barber only shaved the men who did not shave themselves." So that doesn't work either. Often, when the above story is told, one of these last two loopholes is left open. So I had to be careful, when I wrote down the story.

Now we come to a really serious attempt to solve the above puzzle: Maybe there was no barber like the one described in the story. But the story said, "There was once a barber..." So there really was a barber like that, unless the story is a lie! That is the answer, isn't it? The story is a lie. Sorry about that. I told the story of a barber who could not possibly exist. I had good motives. But I guess I told a lie.

Well, our story of the barber is inconsistent. In logic, we don't say that it is a lie. We say that it is inconsistent. "Inconsistent" is much more descriptive, and it is not a sin.

A more Formal form of Barbers Paradox :

Suppose there is a town with just one male barber. In this town, every man keeps himself clean-shaven by doing one of two things:

1. Shaving himself, or
2. going to the barber.

Another way to state this is:

The barber shaves all and only those men in town who do not shave themselves.

All this seems perfectly logical, until we pose the paradoxical question:

Who shaves the barber?

This question results in a paradox because, according to the statement above, he can either be shaven by:

1. himself, or
2. the barber (which happens to be himself).

However, none of these possibilities are valid! This is because:

• If the barber does shave himself, then the barber (himself) must not shave himself.
• If the barber does not shave himself, then the barber (himself) must shave himself.

The above story about the barber is the popular version of Russell's Paradox. The story was originally told by Bertrand Russell. And of course it has a simple solution. It is inconsistent. But the story is not really that simple. The story is a retelling of a problem in set theory.

In set theory, we have sets, collections of objects. These objects may be real physical objects (marbles) or not (cartoon characters, thoughts, or numbers). When we deal with a set, we normally write it down with brackets: {A, B, C}. That set contains three letters, A, B, and C. The set {B,C} is a subset of {A, B, C}. There is a special set with no elements, the empty set {} or ø, as the set of humans bigger than the earth, or the set of odd numbers divisible by two. Some sets contain infinitely many elements, as the set of all even numbers.

A set can contain sets. The set {{A, B, C}, {x, y}} contains two sets {A, B, C} and {x, y}. It also contains the empty set, by the way. All sets contain the empty set. We can define the set of all sets. This set contains {A, B, C} and {{A, B, C}, {x, y}} and every other possible set. Some sets contain themselves. The set of all red marbles does not contain itself, because it contains no sets at all, only marbles. Let's say that S is a set which contains S and {A, B}. Then this is S: {S, {A, B}}. It contains two sets, itself and {A, B}. The set of all sets obviously contains itself. Well, let's construct a very interesting set, the set of all sets which do not contain themselves.

There is something wrong here. Does "the set of all sets which do not contain themselves" sound like "the barber who shaves all men who do not shave themselves?" The story of the barber was inconsistent. The set of all sets which do not contain themselves is inconsistent for the same reason. Does the set of all sets which do not contain themselves actually contain itself, or not? If it contains itself, then it cannot contain itself. If it does not contain itself, then it must contain itself. It is inconsistent.

Now let us come to the notion of what actually Russels Paradox is…..

In the foundation of mathematics Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russellin 1901, showed that the naive set theory created by naive set theory leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.

According to naive set theory(NST), any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox. Symbolically:

 Formal presentation

Define Naive Set Theory (NST) as the theory of predicate logicwith a binary predicate , and the following as axioms:

for all expressions P(x) with just x free

Substitute for P(x). Then by existential instantiation and universal instantiation we have

a contradiction. Therefore NST is inconsistent.