Russell's Paradox

http://plato.stanford.edu/entries/philosophy-mathematics/

The origin of predicativism lies in the work of Russell. On a cue of Poincare,
he arrived at the following diagnosis of the Russell paradox. The argument of
the Russell paradox defines the collection C of all mathematical entities
that satisfy x is an element pf x. The argument then proceeds by asking
whether C itself meets this condition, and derives a contradiction.

The Poincare-Russell diagnosis of this argument states that this definition
does not pick out a collection at all: it is impossible to define a collection
by a condition that implicitly refers to S itself. This is called the vicious
circle principle. Definitions that violate the vicious circle principle are
called impredicative. A sound definition of a collection only refers to
entities that exist independently from the defined collection. Such
definitions are called predicative. As Godel later pointed out, a convinced
platonist would find this line of reasoning unconvincing. If mathematical
collections exist independently of the act of defining, then it is not
immediately clear why there could not be collections that can only be defined
impredicatively (Godel 1944).

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