Barber Paradox

The barber's paradox or the barber 's antinomy is a descriptive variant of Russell's antinomy in logic and set theory , which was established by Bertrand Russell himself in 1918 .

Concept and problem

In 1918 Russell formulated the Barber's Paradox in the following words:

A barber can be defined as one who shaves all those and only those who do not shave themselves.

The question is: does the barber shave himself?

There is a contradiction in trying to answer the question. Because assuming the barber shaves himself, then he is one of those whom he does not shave by definition, which contradicts the assumption. Assuming the opposite is true, and the barber does not shave himself, then contrary to belief, he himself fulfills the quality of those he shaves. Logically, this expresses the following contradicting equivalence for the barber : ${\ displaystyle \, x}$

{\ displaystyle x {\ mbox {shaved}} x \ iff {\ mbox {not}} (x {\ mbox {shaved}} x)}

Russell's solution

Russell said this paradox was easy to resolve. He showed this as early as 1903 in an indirect proof with a variable relation. If you read this backwards, you get a direct proof in which it stands for its variable relation: ${\ displaystyle \, {\ mbox {shaved}}}$

The statement that defines the barber is abbreviated with .

{\ displaystyle x {\ mbox {shaved}} y \ iff {\ mbox {not}} (y ​​{\ mbox {shaved}} y)}

{\ displaystyle \, {\ mbox {B}} xy}

It is the negation of contradiction , that is .

{\ displaystyle x {\ mbox {shaved}} x \ iff {\ mbox {not}} (x {\ mbox {shaved}} x)}

{\ displaystyle {\ mbox {not}} \, {\ mbox {B}} xx}

Therefore, the existential can be introduced: .

{\ displaystyle {\ mbox {There are}} y \ colon {\ mbox {not}} \, {\ mbox {B}} xy}

By introducing the universal quantifier follows: .

{\ displaystyle {\ mbox {For all}} x \ colon {\ mbox {There are}} y \ colon {\ mbox {not}} \, {\ mbox {B}} xy}

By rearranging the quantifiers is finally obtained: .

{\ displaystyle \, {\ mbox {There is no}} x \ colon {\ mbox {For all}} y \ colon {\ mbox {B}} xy}

This provable statement means in plain language: There is no one who shaves exactly those who do not shave themselves. The barber definition, which seems reasonable at first glance, creates a harmless empty term or an empty set. The antinomy takes the barber definition ad absurdum. Russell's solution only shows the definition error, but does not give a solution as to how the barber of a place would be meaningfully defined. That is also unimportant, because his fictional barber definition only served to illustrate his abstract train of thought for any relations. This is the meaning of the barber's paradox. Mathematically and philosophically significant is mainly the variant in which instead of the inverse element predicate, which produces Russell's antinomy , is the most important contradiction in naive set theory . ${\ displaystyle \, {\ mbox {shaved}}}$

variants

There are many variants of the paradox circulating, for example:

The Barber of Seville shaves all men of Seville except those who shave themselves. This embellishment does not provide Russell's pointless definition, it just implies that the barber is not a man from Seville (perhaps a female barber or a barber from the neighboring town working there).

A paradoxical order: “ All mayors are not allowed to live in their own town, but must move to the mayor town of Bümstädt, which has been set up especially for this purpose. Where does the mayor of Bümstädt now live? "

Approaching Russell's Antinomy: A library wants to create a bibliography catalog that lists all bibliography catalogs that do not contain a reference to themselves. Should this catalog also be listed? If so, it receives a reference to itself and yet does not belong in the set of catalogs listed. If not, it contains no reference to itself and yet belongs to this set.

The ancient sophism of Euathlos is also related .

literature

The 42nd story of the solution, Patrick Hughes, George Brecht: Die Scheinwelt des Paradoxons. A commented anthology in words and pictures, title of the Engl. Original edition: Vicious Circles and Infinity , ISBN 3-528-08379-4

Individual evidence

↑ You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? Quoting from: Bertrand Russell: The Philosophy of Logical Atomism , 1918, in: The Collected Papers of Bertrand Russell , 1914-19, Vol 8., p. 228
↑ In this form the contradiction is not very difficult to solve. ibid
^ Bertrand Russell: The principles of mathematics , Cambridge 1903, §102.
↑ with you get exactly the proof for the nonexistence of the Russell class from the above proof scheme. ${\ displaystyle \ ni}$
↑ Duden Unnützes Sprachwissen, C. Hess, 2012.

[1] You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? Quoting from: Bertrand Russell: The Philosophy of Logical Atomism , 1918, in: The Collected Papers of Bertrand Russell , 1914-19, Vol 8., p. 228

[2] In this form the contradiction is not very difficult to solve. ibid

[3] Bertrand Russell: The principles of mathematics , Cambridge 1903, §102.

[4] with you get exactly the proof for the nonexistence of the Russell class from the above proof scheme. ${\ displaystyle \ ni}$

[5] Duden Unnützes Sprachwissen, C. Hess, 2012.