<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. (1995-03-29)

<mathematics> A logical contradiction in set theory discovered by Bertrand Russell. 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. 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 Frankel 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. (2000-11-01)

In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901.Russell, Bertrand, "Correspondence with Frege}.

Russell's teapot is an analogy, formulated by the philosopher Bertrand Russell (1872–1970), to illustrate that the philosophic burden of proof lies upon a person making empirically unfalsifiable claims, rather than shifting the burden of disproof to others.

An attic (sometimes referred to as a loft) is a space found directly below the pitched roof of a house or other building; an attic may also be called a sky parlor or a garret. Because attics fill the space between the ceiling of the top floor of a building and the slanted roof, they are known for being awkwardly shaped spaces with exposed rafters and difficult-to-reach corners.

In classical architecture, the term attic refers to a story or low wall above the cornice of a classical façade. The decoration of the topmost part of a building was particularly important in ancient Greek architecture and this came to be seen as typifying the Attica style, the earliest example known being that of the monument of Thrasyllus in Athens.

(attics)