Russell's Paradox

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.
In logic, some statements are true (Jim is nearsighted), some are false (Jim eats squash). And a collection of statements, such as our story, is either consistent or inconsistent. The following pair of statements is inconsistent:

1. Jim likes vanilla ice cream with Smuckers Plum Jam on it!
2. Jim does not like vanilla ice cream with Smuckers Plum Jam on it.

They contradict one another. They cannot both be true. In fact, one of them is really really false. 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.

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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.
But where did we go wrong? Let's make some lists. A list is a special kind of set. But we know what a list is. A list may be clearer in our minds than a set. We cannot actually physically make infinite lists. But we can certainly define some of them, like the list of all even numbers. So we can deal with infinite lists. We can also make lists of lists. Here is such a list:

1. My shopping list
2. My email address list
3. David Letterman's list of Top Ten Whatevers

This list of lists is real. Now, if we allow infinite lists, then it is no stretch at all to produce the list of all lists, and even the list of all lists which do not contain themselves. And that list is inconsistent.
Well, maybe there are no infinite lists. There are infinite sets, for example, the set of all even numbers. And that is a list: the list of all even numbers. The concept of an infinite list is actually fairly simple.
So we have an inconsistent set. That is not all. We made no mistakes when we constructed the set of all sets which do not contain themselves. And that means that set theory is inconsistent. And that means that logic is inconsistent. And that means that all of mathematics, including algebra and geometry, is inconsistent.
It doesn't invalidate mathematics or logic or set theory. The Pythagorean theorem is still true. But there is some doubt. Kurt Godel (Gödel) proved that Number Theory (and by identical arguments, every branch of mathematics) is inconsistent. He converted Russell's Paradox, the set version, into a statement in Number Theory, and showed that Number Theory is inconsistent. This had huge repercussions in the world of mathematics. All of this leads to the following problem:

1. There are things that are true in mathematics (based on basic assumptions).
2. There are things that are false.
3. There are things that are true that can never be proved.
4. There are things that are false that can never be disproved.

And that is a problem, because we cannot ever tell if something is true unless we can prove it.