Definition
Russell's Antimony
Let
be the set of all sets. Every element of the power set is, by definition, a subset of . Since every subset of is itself a set, it must, by definition of , be an element of . Therefore, we can conclude that Based on that, we have found a surjective function
defined as follows:
for all that are subsets of - To those elements of
that are not subsets of , we assign the empty set , which is guaranteed to be an element of . This appears to be a contradiction as Cantor’s Theorem states no set can be mapped surjectively onto its power set.
Examine the first set
further: This constitutes Russell’s Paradox. The collection of all sets that do not contain themselves as an element lead to a direct contradiction when we ask: Does this set
contain itself, or not?.
By definition, an object
is an element of iff satisfies . Since this condition must hold for every
in the domain , it must specifically hold for the case where : R \in R \iff R \not\in R