Definitions

# explaining away

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.

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. (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.) 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 of word and then lexicographically (in dictionary order).

Now, we may map each definition to the set of cardinal numbers, 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, i.e. the number of letters in the definition equals the integer. If, for example, the 43 letters long (ignoring the spaces) description "not divisible by any integer other than 1 and itself" were assigned to the number 43, 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: "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. Finally, the paradox becomes: Is n Richardian? Suppose n is Richardian. This is only possible if n does not have the property designated by the defining expression which n is correlated with. In other words, this means n is not Richardian, contradicting our assumption. However, if we suppose n 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 "n is Richardian" can not consistently be designated as either true or false.

Richard's Paradox is fallacious. 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. 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.

Explaining away requires distinguishing 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.