Russell's Paradox
Let denote the set R of all sets S that do not belong to themselves. The inconsistency of the existence of this set is known as Russell's paradox.
Solutions to the paradox restrict the notion of set construction in some way. To illustrate this in terms of our notation, let X = {x ∈ A : P(x)} denote the set of every element of A satisfying the predicate P(x). The canonical restriction on set builder notation asserts that X is a set only if A is already known to be a set. This restriction is codified in the axiom schema of separation present in standard axiomatic set theory. Note that this axiom schema excludes R from sethood.
Read more about this topic: Set-builder Notation
Famous quotes containing the words russell and/or paradox:
“Sincerity is impossible, unless it pervade the whole being, and the pretence of it saps the very foundation of character.”
—James Russell Lowell (1819–1891)
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)