Editing Richard's paradox (section)

== Variation: Richardian numbers ==

A variation of the paradox uses integers instead of real numbers, while preserving the self-referential character of the original. Consider a language (such as English) in which the [[arithmetic|arithmetical properties]] of integers are defined. For example, "the first natural number" defines the property of being the first natural number, one; and "divisible by exactly two natural numbers" defines the property of being a [[prime number]] (It is clear that some properties cannot be defined explicitly, since every [[deductive system]] must start with some [[axiom]]s. 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). 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, first by length and then [[Lexicographical order|lexicographically]].

Now, we may [[map (mathematics)|map]] each definition to the set of [[natural number]]s, such that the definition with the smallest number of characters and alphabetical order will correspond to the number 1, the next definition in the series will correspond to 2, and so on. 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" happened to be 43rd, then this would be true. Since 43 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. However, this may not always be the case. If the definition: "divisible by 3" were assigned to the number 58, then the number of the definition does ''not'' have the property of the definition itself, since 58 is itself not divisible by 3. 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".) Thus in this example, 58 is Richardian, but 43 is not. 

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''. For example, the definition "being Richardian" might be assigned to the number 92. Finally, the paradox becomes: Is 92 Richardian? Suppose 92 is Richardian. This is only possible if 92 does not have the property designated by the defining expression which it is correlated with. In other words, this means 92 is not Richardian, contradicting our assumption. However, if we suppose 92 is not Richardian, then it does have the defining property which it corresponds to. This, by definition, means that it is Richardian, again contrary to assumption. Thus, the statement "92 is Richardian" cannot consistently be designated as either true or false.