# Russell Bertrand Arthur William

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# Russell's Attic

<mathematics> An imaginary room containing countably many pairs of shoes (i.e. a pair for each natural number), and countably many pairs of socks. How many shoes are there? Answer: countably many (map the left shoes to even numbers and the right shoes to odd numbers, say). How many socks are there? Also countably many, we want to say, but we can't prove it without the Axiom of Choice, because in each pair, the socks are indistinguishable (there's no such thing as a left sock). Although for any single pair it is easy to select one, we cannot specify a general method for doing this.

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<mathematics> A logical contradiction in set theory discovered by the British mathematician Bertrand Russell (1872-1970). If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa. This contradiction infects set theory when it is permissible to speak of "all sets" or set complements without qualification, or when a set is defined loosely as any collection of any elements, or when every predicate (intension) determines a set (extension). See complement The paradox stems from the acceptance of the following axiom: If P(x) is a property then

```	{x : P}

```
is a set. This is the Axiom of Comprehension (actually an axiom schema). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent.

In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself:

```	r = \ x . not (x x)

```
If we now apply r to itself,

```	r r = (\ x . not (x x)) (\ x . not (x x))
= not ((\ x . not (x x))(\ x . not (x x)))
= not (r r)

```
So if (r r) is true then it is false and vice versa.

An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of set theory suggest the existence of the paradoxical set R.

Zermelo Fr"nkel set theory is one "solution" to this paradox. Another, type theory, restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself.

A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway.

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# Russell's theory of descriptions

<logic, philosophy of language> roughly, the view that sentences in which phrases of the form the-so-and-so appear can be reduced to more revealing logical forms in which "the" disappears and in which there is no longer any temptation to think that such phrases are like proper names. E.g. "The present king of France is bald" becomes "There exists something which is presently kind of France and there is no other individual who is such and that individual is bald". Russell's theory has been called a paradigm of philosophy.

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