Richard's paradox

Richard's paradox is a falsidical 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.

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.

Related Topics:
Arithmetical properties - Prime number

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(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.)

Related Topics:
Deductive - Axiom

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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).

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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.

Related Topics:
Map - Cardinal number

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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.

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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.

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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".)

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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?

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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.

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