Richard's paradox is a fallacious paradox of mathematical mapping first described by the French mathematician Jules Richard in 1905. Today, it is ordinarily used in order to show the importance of carefully distinguishing between mathematics and metamathematics. Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... This article is in need of attention from an expert on the subject. ... Jules Antoine Richard (1862-1956) was a French mathematician. ... 1905 (MCMV) was a common year starting on Sunday (see link for calendar). ... Euclid, detail from The School of Athens by Raphael. ... Metamathematics is mathematics used to study mathematics. ...

Description of the paradox

Consider a language (such as English) in which the arithmetical properties of integers are defined. For example, "the first natural number" defines the property of being the first natural number, one; and "not divisible by any integer other than 1 and itself" defines the property of being a prime number. Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ... A prime number (or a prime) is a natural number that only has trivial divisors. ...

(It is clear that some properties cannot be defined explicitly, since every deductive system must start with some axioms. But for the purposes of this argument, it is assumed that phrases such as "an integer is the sum of two integers" are already understood.) In traditional Aristotelian logic, deductive reasoning is inference in which the conclusion is of no greater generality than the premises, as opposed to abductive and inductive reasoning, where the conclusion is of greater generality than the premises. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...

While the list of all such possible definitions is itself infinite, it is easily seen that each individual definition is composed of a finite number of words, and therefore also a finite number of characters. Since this is true, we can order the definitions lexicographically (in dictionary order). In mathematics, the lexicographical order, or dictionary order, is a natural order structure of the cartesian product of two ordered sets. ...

Now, we may map each definition to the set of cardinal numbers, such that the definition with the smallest number of characters and alpabetical order will correspond to the number 1, the next definition in the series will correspond to 2, and so on. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...

Since each definition is associated with a unique integer, then it is possible that occasionally the integer assigned to a definition fits that definition. If, for example, the definition: "not divisible by any integer other than 1 and itself" were assigned to the number 23, then this would be true. Since 23 is itself not divisible by any integer other than 1 and itself, then the number of this definition has the property of the definition itself.

But this will not always be the case. If the definition: "the first natural number" were assigned to the number 4, then the number of the definition does not have the property of the definition itself.

This latter example will be termed as having the property of being Richardian. Thus, if a number is Richardian, then the definition corresponding to that number is a property that the number itself does not have.
(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".)

Now, since the property of being Richardian is itself a numerical property of integers, it belongs in the list of all definitions of properties. Therefore, the property of being Richardian is assigned some integer, n. And finally, the paradox: Is n Richardian?

Consider n as Richardian. This is only possible if n does not have the property designated by the defining expression which n is corellated with. In other words, it is only true if n is not Richardian. Thus, n is Richardian if n is not Richardian. The statement "n is Richardian" is both true and false.

Resolving the paradox

Richard's Paradox is fallacious; it is but a magic trick, and can be easily explained away. An essential but tacit assumption concerning the ordering of definitions was ignored while setting up the paradox.

It was agreed to consider the arithmetical properties of integers, i.e., properties that can be spoken about using additions, multiplication, etc. But then later in the paradox a definition was added to the series which involves reference to the notation used in arithmetical properties. This is obviously not allowed. The definition of being Richardian does not belong to the series initially intended, because this definition involves meta-mathematical notions such as the number of letters occurring in expressions. Metamathematics is mathematics used to study mathematics. ...

Explaining away Richard's Paradox is as easy as being careful to distinguish between statements within arithmetic (which make no reference to any system of notation) and statements about some system of notation in which arithmetic is codified.

See also

• Berry paradox, which also uses numbers definable by language.
• algorithmic information theory
• Gödel's proof

The Berry paradox is the apparent contradiction that arises from expressions such as the following: The smallest positive integer not nameable in under eleven words. ... In computer science, algorithmic information theory is a field of study which attempts to define the complexity (aka descriptive complexity, Kolmogorov complexity, Kolmogorov-Chaitin complexity, or algorithmic entropy) of a string as the length of the shortest binary program which outputs that string. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...

References

• Jules Richard, "Les Principes des mathématiques et le problème des ensembles", Revue générale des sciences pures et appliquées (1905); translated in Heijenoort J. van (ed.), Source Book in Mathematical Logic 1879-1931 (Cambridge, Mass., 1964).