Russell’s Paradox

Lecture 24

Section 5.4

Fri, Mar 10, 2006

Russell’s Paradox

Let A be any set that you have ever seen. Then, most likely, A A. For example

A set of integers is not itself an integer.A set of rectangles is not itself a rectangle.A set of points in the plane is not itself a

point in the plane. Is it possible that A A for some set A?

Russell’s Paradox

Let S = {A | A A}. Let P(x) be the predicate “x x.” Is S S?

If S S, then • S satisfies the predicate.• So P(S) is true.• But P(S) says that S S.

Russell’s Paradox

If S S, then• S does not satisfy the predicate.• So P(S) is false• But that means that S S.

Therefore, “S S” is neither true nor false. This is a paradox.

The Barber Paradox

In a certain town, there is a barber who cuts the hair of every person (them and only them) in the town who does not cut his own hair.

Question: Who cuts the barber’s hair?

The Bibliography Paradox

An author writes a book about bibliographies.

He decides to list in the bibliography of this book all books that do not list themselves in their own bibliographies.

Should he list his own book?

The Title Paradox

Now the author decides to title his book

The Title ofof this Book

Contains TwoErrors

An Interesting “Theorem”

Theorem: This theorem has no proof. Can you prove this theorem? Is this theorem true?

If this theorem were false, then it would have a proof.

But you can’t prove a false theorem.Therefore, it must be true.

But doesn’t that argument constitute a proof that the theorem is true?

The Berry Paradox

Consider the set A of all positive integers that can be described using fifty English words or less.“one”“the square of eleven”“the millionth prime”“the millionth prime times the billionth

prime, plus ten”

The Berry Paradox

Let B = N – A. That is, B is the set of all positive integers

that cannot be described using fifty English words or less.

B is not empty. (Why?) What is the smallest number in B? It is called the Berry number, after G. G.

Berry, an Oxford University librarian.

The Berry Paradox

Whatever the Berry number is, it is “the smallest positive integer that cannot be described using fifty English words or less.”

But that description itself uses less than 50 English words and it describes that number.

See The Berry Paradox.