The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics.
Kurt Gödel specifically cites Richard's antinomy as a semantical analogue to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I".
The original statement of the paradox, due to Richard (1905), is strongly related to Cantor's diagonal argument on the uncountability of the set of real numbers.
Richard's paradox results in an untenable contradiction, which must be analyzed to find an error.
Thus the resolution of Richard's paradox is that there is not any way to unambiguously determine exactly which English sentences are definitions of real numbers (see Good 1966).
For, if it were possible to define this set, it would be possible to diagonalize over it to produce a new definition of a real number, following the outline of Richard's paradox above.
A variation of the paradox uses integers instead of real numbers, while preserving the self-referential character of the original.
(More formally, "x is Richardian" is equivalent to "x does not have the property designated by the defining expression with which x is correlated in the serially ordered set of definitions".)
Set theories such as ZFC are not based on this sort of predicative framework, and allow impredicative definitions.
They believe the definition of r is invalid because there is no well-defined notion of when an English phrase defines a real number, and so there is no unambiguous way to construct the sequence rn.
Although Richard's solution to the paradox did not gain favor with mathematicians, predicativism is an important part of the study of the foundations of mathematics.
Predicativism was first studied in detail by Hermann Weyl in Das Kontinuum, wherein he showed that much of elementary real analysis can be conducted in a predicative manner starting with only the natural numbers.
More recently, predicativism has been studied by Solomon Feferman, who has used proof theory to explore the relationship between predicative and impredicative systems.