## A close shave for set theory

Suppose you walk past a barber's shop one day, and see a sign that says

"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."

This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself? If he does, then he mustn't, because he doesn't shave men who shave themselves, but then he doesn't, so he must, because he shaves every man who doesn't shave himself... and so on. Both possibilities lead to a contradiction.

This is the Barber's Paradox, discovered by mathematician, philosopher and conscientious objector Bertrand Russell, at the begining of the twentieth century. As stated, it seems simple, and you might think a little thought should show you the way around it. At worst, you can just say "Well, the barber's condition doesn't work! He's just going to have to decide who to shave in some different way." But in fact, restated in terms of so-called "naïve" set theory, the Barber's paradox exposed a huge problem, and changed the entire direction of twentieth century mathematics.

In naïve set theory, a set is just a collection of objects that satisfy some condition. Any clearly phrased condition is thought to define a set - namely, those things that satisfy the condition. Here are some sets:

- The set of all red motorcycles ;
- The set of all integers greater than zero;
- The set of all blue bananas - which is just the empty set!

This set is not a member of itself

Some sets are not members of themselves - for example, the set of all red motorcycles - and some sets are - for example, the set of all non-motorcycles. Now what about the set of all sets which are not members of themselves? Is it a member of itself or not? If it is, then it isn't, and if it isn't, then it is... Just like the barber who shaves himself, but mustn't, and therefore doesn't, and so must!

So now we realise that Russell's Barber's Paradox means that there is a contradiction at the heart of naïve set theory. That is, there is a statement S such that both itself and its negation (not S) are true. The particular statement here is "the set of all sets which are not members of themselves contains itself". But once you have a contradiction, you can prove anything you like, just using the rules of logical deduction! This is how it goes.

- If S is true, and Q is any other statement, then "S or Q" is clearly true.
- Since "not S" is also true, so is "S or Q and not S".
- Therefore Q is true, no matter what it is.

The paradox raises the frightening prospect that the whole of mathematics is based on shaky foundations, and that no proof can be trusted. In essence, the problem was that in naïve set theory, it was assumed that any coherent condition could be used to determine a set. In the Barber's Paradox, the condition is "shaves himself", but the set of all men who shave themselves can't be constructed, even though the condition seems straightforward enough - because we can't decide whether the barber should be in or out of the set. Both lead to contradictions.

Attempts to find ways around the paradox have centred on restricting the sorts of sets that are allowed. Russell himself proposed a "Theory of Types" in which sentences were arranged hierarchically. At the lowest level are sentences about individuals. At the next level are sentences about sets of individuals; at the next level, sentences about sets of sets of individuals, and so on. This avoids the possibility of having to talk about the set of all sets that are not members of themselves, because the two parts of the sentence are of different types - that is, at different levels.

But to be a satisfactory philosophy, we have to be able to say why you are not allowed to mix levels. Although, for example, it is not true that the property of being red is itself red, this is surely a wrong statement, rather than actually meaningless. And there are properties that seem reasonably to apply to themselves - the Theory of Types disallows statements such as "It's nice to be nice" but really this seems like a reasonable and true statement!

For this and other reasons, the most favoured escape from Russell's Paradox is the so-called Zermelo-Fraenkel axiomatisation of set theory. This axiomatisation restricts the assumption of naïve set theory - that, given a condition, you can always make a set by collecting exactly the objects satisfying the condition. Instead, you start with individual entities, make sets out of them, and work upwards. This means you do not have to suppose that there is a set of all sets, which means you don't have to try to divide that set up into those sets that contain themselves and those which don't. You only have to be able to make this division for the elements of any given set, which you have built up from individual entities via some number of steps.

To end on a more flippant note, if Russell had been aware of the inbuilt sexism of the language of his day, the course of twentieth century mathematics might have been different. There is an easy solution to the Barber's Paradox, which doesn't require the opening of any nasty cans of set-theoretic worms. Just make the barber a woman...

## About the author

**Helen Joyce** is editor of *Plus*.

## Comments

## Barbershop quintet

There are in fact at least five barbers, each of whom shaves himself if and only if he doesn't.

The most famous one, call him Bertrand, shaves all and only those who don't shave themselves.

Alfred, on the other hand, is shaved by all and only those who don't shave themselves.

Gottlob shaves all and only those who don't shave him.

Georg is shaved by all and only those whom he doesn't shave.

Augustus (my favourite) shaves all and only those whom he doesn't shave.

Chris G

## Round and round

The principal of Russell College instructed her maintenance staff "If the front door is red, then paint it any other colour. But if it's any other colour, then paint it red."

That was some years ago and they're still painting.

## Problem of the Paradox

The real problem with the paradox is in the verb shave is used differently in two instances.

To demonstrate this, here is my solution to the paradox:

Those men that shave themselves do not go to the barber.

Those men that do not shave themselves are shaved by the barber. This should really be, those men that do not shave themselves are barbered by the barber.

And like that, the paradox is solved.

Does the barber shave himself? Yes, but he does not barber himself.

Alex

## A contradiction means a premise is false

The barber could be a woman, a pre-beard child, an American Indian, skin can't grow hair due to fire accident on the face making it impossible, etc. Essencially, any situation where a barber growing a beard doesn't apply.

Let's assume the barber can grow a beard (some women included), and shaving includes all methods of removing hair (including things like fire to the face), then the solution is still obvious. One of premises in the original statement is false: "Do you shave yourself? If not, come in and I'll shave you! I shave [everyone] who does not shave himself, and noone else."

One of these premises are false:

1. Everyone in town has a clean shave, which includes the barbor. (so the barbor has a beard)

2. "I shave [everyone] who does not shave himself..." (so he doesn't shave everyone who does not shave himself)

3. "...and noone else." (so he shaves others too)

Bottom line: the orignal statement has to be false.

## solution or not?

If the barber were to wax his beard or hair off or have it removed by plucking or in any other way that does not include a razor, would that count as shaving? If not, then that would save him and everyone else the trouble of deciding whether or not he should shave himself.

## This Awesome Theory!

To Helen Joyce,

Yes, this is all well but what actually is the answer? I would like to know!

Then also do you know any sites (other that this one) that give you long lost/forgotten mathematical mysteries that DON'T involve any algebra?

Please if you go post it on this site! Also I do think this is a bit sexist so I think it's best if you kind of think through what you're going to type and the viewpoints/opinions you're going to get about it. I thank you for your time reading this.

Yours Mysteriously,

Anonymous

## Hidden sexism?

Clearly, the "anti-sexist" way-out for the barber's paradox could have worked - had the original statement been that the barber is the person who shaves any MAN that does not shave himself. But it doesn't! It says that the barber shaves anyONE that doesn't shave himself - that is, women included.

To add to the "God paradox": can God create a barber? And must that barber then be of a non-human race?

## Just Letting You Know

I used your Barber paradox in my blog today and included a link to the page I used. Great website, I hope I didn't anger you by using the example.

The Green Dude

## A close shave for set theory...

"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and no one else."

"Does the barber

shavehimself? If he does, then he mustn't, because he doesn't shave men who shave themselves, but then he doesn't, so he must, because he shaves every man who doesn't shave himself... and so on. Both possibilities lead to a contradiction. "A seemingly true paradox, but really just a

naive question.The parameters of the question are undefined.For instance; what defines a "shave"?Is it considered a shave when you:

A.Remove all the hair from your face?B.The initial cutting of hair that defines a shave?Lets define this by way of common language:

I can start to shave and stop to answer the phone,

most likely my wife reminding me to shave,

so I answer,

"I am not done shaving but I was shaving"So by the aforementioned conversation, the "act of shaving" is the term by which we agree to acknowledge what consitutes a shave. There for by this "definition" the answer would have to be;

He would start to shave himself but never finish and from the point on only get some one else to shave him...## The different identical objects or The uniqueness of truth

A perfectly true sentence: "Identical objects are different."

To make this statement interesting i will give the answer two weeks after it's publication.

## The Barber's Paradox

The aim of every science is to discover the laws that could explain one or another phenomenon. Once these laws are discovered, then science proceed to study the other phenomena, which in the nature are of an infinite set. It is interesting to note that in the process of discovering a law, for example in physics, people make thousands of experiments, build proves, among them some experience, or evidence - useful for understanding a certain phenomenon, but other experiments or evidence proved fruitless. But it is found only with hindsight, when the law is already discovered. Therefore, with the discovering of the law it is enough to show 2 - 3 experiments or prove to verify its correctness. All other experiments were the ways of study and there is no need to repeat them, to understand how the law works. In the exact sciences, it is understood, and therefore the students studies only the information that is necessary to understand specific phenomena. By no means is the case with the study of formal logic.

Formal logic, as opposed to other sciences: physics, chemistry, mathematics, biology and so on, studies not an infinite number of phenomena in nature, but only one how a man thinks, how he learns the world surrounding us, and how people understand each other. In other words, what laws govern the logic of our thinking, i.e., our reasoning and judgments in any science or in everyday life.

By the beginning of XVIII century four laws of logic were formulated :the law of identity, Law of Contradiction, the Law of Excluded Middle and the Law of Sufficient Ground. The first three laws were formulated by Aristotle in the 4 th century BC, as the 4th law was introduced by Leibniz at the beginning of 18 century. So far for more than 300 years, none of the philosophers discovered neither the 5th or 6 th law of formal logic. During this time, all the "discovery" of formal logic were limited only to it’s the distortion and confusion.If to consider a formal logic of Aristotle from the point of view of its essence, then its center of gravity is its Laws, that were discovered by Aristotle, based on analysis of the different types of syllogism, which Aristotle classified to track down those Laws. In his research, the syllogisms played the same role as the experiments in physics or chemistry for the discovery of regularities, to explain the process of certain events. Once these logical laws of thinking have been discovered,the syllogisms have fulfilled their role. And it would be foolish to assume that our knowledge in any science, is built only on Aristotle's syllogisms or others discovered later. Whatever syllogisms would not have been discovered since Aristotle, none of them had added something new in the laws of formal logic revealed by Aristotle and Leibniz. But philosophers still continue to analyze Aristotle's syllogisms, a historic mission of which ended more than 2000 years ago. Moreover, after the discovering of 4th of law of formal logic, the law of sufficient ground , the legality of any syllogism is easy checked from the viewpoint of the four laws of formal logic, because all our judgments and inferences must be obeyed to these laws, to be true.

Bertrand Russell is the one who belongs to this category of the philosophers, who in his book "History of Western Philosophy", examining the formal logic of Aristotle, has continued to pick weaknesses in his syllogisms,rather than focus his attention on the importance of the laws of formal logic in the human knowledge and to point out to the incompleteness of their definitions.Here he writes about the formal logic: "Aristotle's most important work in logic is the doctrine of the syllogism... Apart from such inferences as the above, Aristotle and his followers thought that all deductive inference, when strictly stated, is syllogistic. By setting forth all the valid kinds of syllogism, and setting out any suggested argument in syllogistic form, it should therefore be possible to avoid fallacies.This system was the beginning of formal logic, and , as such, was both important and admirable. But considered as the end, not the beginning, of formal logic, it is open to three kinds of criticism:(1) Formal defects within the system itself. (2) Over-estimation of the syllogism, as compared to other forms of deductive argument. (3) Over-estimation of deduct5ion as a form of argument." (Bertrand Russell, "A History of Western Philosophy", p196-197, published by Simon&Schuster)

As we see, in his chapter "Aristotle's Logic" he did not mention at all about the importance of three laws of formal logic discovered by Aristotle. As I said earlier, the power of formal logic, its common to all sciences, based on its four laws, rather than on different types of syllogism. Russell's misunderstanding of this fact led him to an underestimation and distortion of formal logic.

Such a perversion, and was introduced by the famous philosopher, B. Russell, in formal logic, as shown by his following explanation to the 3rd position above: " All the important inferences outside logic and pure mathematics are inductive, not deductive; the only exceptions are law and theology, each of which derives its first principles from an unquestionable text, viz, the statute books or the scriptures" (p. 199)

And in another place he writes:"Valid syllogisms, in fact, are only some among valid deductions, and have no logical priority over others. The attempt to give pre-eminence to the syllogism in deduction misled philosophers as to the nature of mathematical reasoning. Kant, who perceived that mathematics is not syllogistic, inferred that it uses extra-logical principles, which, however, he supposed to be as certain as those of logic. He, like his predecessors, though in a different way, was misled by respect for Aristotle" (p.199)

Rather than to say that even outside of logic and pure mathematics, four of law of formal logic, without any doubt remained valid,he emphasizes on the syllogisms that do not constitute the essence of formal logic and for this reason are not general to all science. His misunderstanding of role of the laws of formal logic in human thinking is confirmed by his paradox with the notion of set. Let us show how this paradox violates the basic laws of formal logic.The paradox of Russell in the original formis linked to the notion of set or class. But the whole world knows it in another interpretation. Russell proposed the following popular version of discovered him paradox of set theory. Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves. Under this scenario, we can ask the following question: Does the barber shave himself ? Asking this, however, we discover that the situation presented is in fact impossible:If the barber does not shave himself, he must abide by the rule and shave himself.

If he does shave himself, according to the rule he will not shave himself.

In this paradox, all the men in town are divided into two categories : those who shave themselves, and those who do not shave themselves. And barber is in one of these categories (one sufficient ground) , as he is a man from the town. But on the other hand, the man identified as a barber (another sufficient ground) with the functions that are contrary to the first two categories or sufficient ground:

he shaves all and only those men in town who do not shave themselves. Thus barber is determined in two ways for both categories:(1) as a man who shave himself and (2) as a barber, who shave all and only those men in town who do not shave themselves; or (1) as a man who does not shave himself and (1) as a barber, who shave all and only those men in town who do not shave themselves. The paradox violated the laws of formal logic: the law of identity and the law of sufficient ground. Violation of the Law of Identity takes place by introducing into the paradox of the two sufficient grounds: town men and town barber. And if our assumptions have violated the basic laws of formal logic, the conclusions would be incorrect.

For example, imagine a foreigner with an excellent memory, who memorized 5000 English words, but he absolutely does not know the rules of the English language. Rather than to say: "Today I read an interesting book", he said: "I book today read an interesting ." This sentence does not make sense to anybody because it has no meaning. It is well known that in order to learn new language one has to know besides the words, all the rules of language, another words to be familiar with the laws of the language which one tries to learn. The same thing is in any science, including formal logic. But it does not mean that the English language does not always work or it is not powerful enough to express any idea, because the foreigner breaks the rule of the language. But based on the same logic philosophers come to the conclusion that the formal logic is not perfect , when confronted with logical paradoxes that violating the laws of formal logic .

It is interesting to note that Russell is aware of the presence in the paradox of two sets of definitions, which poses a problem. Russell's own answer to the puzzle came in the form of a "theory of types." The problem in the paradox, he reasoned, is that we are confusing a description of sets of numbers with a description of sets of sets of numbers. " - but he does not associate it with a violation of the law of Identity , not to mention about the law of Sufficient ground, because in this case there would be nothing to talk about and there would be nothing new to be discovered, because this problem has been resolved by Aristotle over 2000 years ago.His critique of Aristotelean formal logic remind me the story about the curious man who visited Zoo and tried to pay attention to everything including tiny insects, but after his visit of Zoo, he started to boast about his knowledge of animals that he saw, and one of his listeners asked him did he see an elephant in the Zoo and he said that he did not noticed one.

As I have said early, to understand the importance of the law of sufficient ground in its aggregation it is necessary to understand very well the relationship of formal and dialectical logic, and Russell denied the existence of the latter: "Even if (as I myself believe) almost all of the teachings of Hegel is false, it is still important, which not only belongs to history, as it best represents a certain kind of philosophy, which have other, less coordinated and less than comprehensive." After this statement about the dialectical logic, and with such understanding of formal logic, he can not be called not only as a great philosopher but even as a philosopher, because he did not introduce anything new to the development of formal logic of Aristotle, except of the distortion and perversion....

Ilya Stavinsky

## The God Paradox

Can God create a stone that he can not lift up?

This is based on the general assumption that everything is possible to god.So if he creates a stone which he cann't lift up,then it breaks the condition he can lift anything.At the same time if he cann't create a stone, then its against that he can create anything.

## Can God create a stone that he can not lift up?

Well God can create a stone, that is already lifted up, since it is already lifted up, he cannot lift it up again.

condition proved.

## can god create a stone he can not lift?

if god can't lift up a stone that is already lifted, it states that god can't do anything, which he can, thus he can lift an already lifted stone.

## The God paradox

An axiom: God is infinite. (True)

So let's study the action of creating that stone which God cannot lift up.

The stone would have to be of an infinite mass.

Therefore when God creates that stone, He would no longer exist.

Why? If God would still exist outside that stone, that would mean that the stone would be of a finite mass, which is false.

Therefore God would no longer exist while the stone which He cannot lift exists. So there would be no one left to lift that stone.

That proves that the paradox does not take place.

## The God Paradox

Hmm well your axiom

God is infinite..

violates your whole proof so nice try but it's a definite no go..

axiom : God is Infinite

predicate : God is Inifinite If and only if God has no mass

otherwise God would take up all mass and there would be no stones.. :) (this was your argument not mine.. that a stone can't be infinite mass or god would be gone. , which is contrapositive to this this proposition)

nice try though..

So.

if god is infinite , he sure must be massless, energyless too for that matter..

## The Barber's Paradox

This is the first time I've seen this paradox. It is confusing. But it just seems like a play on words. You can't seem to cross more than two levels unless you change the definition of the set. If A=odd #s and B=even #'s then A and B are not members of themselves. So if C=A+B then C could be considered the set of sets that are not members of themselves, or C could be the set of all real integers. But if A was the whole number digits of pi and B was all prime #'s then you can't really define A+B. I guess it is in the hierarchy. You can't really cross two levels by defining the second level as C when it is really undefined. You're just puting a label on something that doesn't fit. It is like saying A is all red motorcycles and B is all blue motorcycles and calling the set A+B the set of motorcycles. It's not true. You either would have to add them together and say the set of all blue and red motorcycles, or keep it as 2 separate sets that can not be combined because there is no definition to that set. So by saying you have a set of sets that are not members of themselves, you could be talking about a set of infinite possibilities with no real parameter defined. I guess in my opinion when you say a set of sets that are not a member of themselves, you could be talking about anything. Now if that "anything" happens to be a group of things that have a physical rule that each individual thing shares then it could be a set defined by that rule. But, if there is no physical common bond other than the "anything" is just a group of sets drawn together and labeled with a variable, then in my opinion they are not really a set. More of an irrational set. And in the Barber Paradox it seems to me that this is just an irrational statement. I shave everyone that doesn't shave themselves. Well, he misspoke. He shaves everyone who doesn't shave themselves, other than himself. The set was not defined properly. Basically, the barber is the definition of the set. The flaw to me is this. People who do not shave themselves are individuals with a rule. The barber is really a parameter, not an individual. You must first be an individual then follow the rule to be in the group. How can a set be a member of individual motorcycles that are red? It's not a motorcycle, it is a set. Anyway, that's my thoughts on the subject. I guess I would have to see how this applies in mathematics to fully understand.