Tuesday, November 2, 2010

Russell's Paradox

In 1901 Russell discovers his famous mathematical paradox.

Informal presentation

Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal".

Now we consider the set of all normal sets, R. Attempting to determine whether R is normal or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if it were abnormal, it would not be contained in the set of normal sets (itself), and therefore be normal. This leads to the conclusion that R is both normal and abnormal: Russell's paradox.

Russell-like paradoxes

As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:

* A transitive verb , that can be applied to its substantive form.

Form the sentence:

The er that s all (and only those) who don't themselves,

Sometimes the "all" is replaced by "all ers".

An example would be "paint":

The painter that paints all (and only those) that don't paint themselves.

or "elect"

The elector (representative), that elects all that don't elect themselves.

Paradoxes that fall in this scheme include:

* The barber with "shave".
* The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves.
* The Grelling–Nelson paradox with "describer": The describer (word) that describes all words, that don't describe themselves.
* Richard's paradox with "denote": The denoter (number) that denotes all denoters (numbers) that don't denote themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all denoters (numbers) that don't denote themselves" is here called Richardian.)

(Grpahic comic from: Logicomix: An Epic Search for Truth (Amazon))

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