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Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

Richard’s Paradox

Page last updated 28 Feb 2023


Richard’s paradox is as follows: (Footnote: Richard, Jules (1905). Les Principes des Mathématiques et le Problème des Ensembles (The principles of mathematics and the problem of sets).)


English consists of various symbols - 26 letters, spaces, and various punctuation marks.


All permutations of the symbols of any given alphabet can be put in a dictionary style list, by first deciding an alphabetical order of the symbols, then the first part of the list will be that list of single symbols, then taking all permutations of two symbols in an alphabetical order, then taking all permutations of three symbols in alphabetical order, and continue the list by taking all permutations of four symbols, and so on. The permutations may contain the same symbol repeated several times. This is called a lexicographical order.


For any integer n, any permutation of the symbols taken n at a time will be in the list; and, since everything that can be written with finitely many words is a permutation of the symbols, everything that can be written in English will be in the list given by the above definition. Some of the permutations in the list will be definitions of numbers. We ignore all permutations that are not definitions of numbers, and we make a second list, by assigning 1 to the first definition of a number, 2 to the second definition of a number, 3 to the third definition of a number, and so on. We now have a list of all numbers that are defined in English by finitely many words, which we call the list E.


Given this list, we can now define a number N not belonging to this list:

“Let p be the digit in the nth decimal place of the nth number of the list E. We define a number beginning with 0. and its nth decimal place is p + 1 if p is not 8 or 9, and 1 otherwise.”


This number N does not belong to the list. If it were the nth number in the list, the digit at its nth decimal place would be the same as the one in the nth decimal place of that number, which is not the case. But the number N is defined by the above definition within the quotes above, which is a permutation of the symbols of English. This is the contradiction that is the paradox.


Richard actually goes on to say that there is only an apparent contradiction, as follows:

“The collection of symbols within the quotes, which we will call G, is a permutation that will appear in the list E. But it is defined in terms of the set E, which has not yet been defined. Hence it will not appear in the list E. G has a defined meaning only if the list E is totally defined.”


Here Richard hits the nail on the head, and points out that the supposed paradox is dependent on the ambiguity of the definitions involved. But it seems that almost everyone prefers to ignore this fact and instead believe that there is something significant in it.


And if you try to actually provide a clear definition of the list, you will need to clarify the vague definition above. The list is defined in terms of a list B that includes every English expression, then if that list B could be expressed in English, then it would also include the English expression that defines that list B itself. But that is not possible - the definition of the list B requires that there is a free variable x in the definition, (Footnote: The free variable has the domain of all natural numbers, so it can take the value of any natural number from 1 upwards.) so if there could be a definition L in the list that actually is the definition of the list B, then that definition L would be given by the substitution of some specific number n for x in the definition. But that then would mean that the definition L cannot have any free variable - which would mean that it could not actually be the definition of the list B.


Since the list B cannot be defined within the English language, then the list E of all English expressions for natural numbers cannot be defined in English either, since the list E of all English expressions for natural numbers is defined in terms of the list B. Hence the argument that there is a number N that is not in the list E fails, since it assumes that the definition of N is possible as an English expression - but that is not possible, since it is defined in terms of the list E, which is defined in terms of the list B, which cannot be defined in English.


This is all blindingly obvious if one at all familiar with the notions of sub-language and meta-language, since the list that can list all symbol sequences of a given language must be in a language that is a meta-language to that given language, and cannot be in the given language itself. For more details, see Non-Diagonal Proofs: Enumerations in different language systems.


Richardian numbers

A variation of the paradox uses natural numbers instead of real numbers, and supposes a list of all English expressions that describe properties of natural numbers.


It relies on the claim:

An expression that is the nth expression in the list can describe a property that may or may not apply to the nth number in the list.

and from that claim the assertion:

If the nth expression describes a property that does not apply to the number n, then the number is said to be Richardian.


The paradox arises as follows:

Since the property of being Richardian is itself a numerical property of integers, it must be included in the list of all definitions of properties. Therefore, the property of being Richardian is the nth expression in the list for some number n.

Is n Richardian? If n is Richardian, then that defines n as not having the property described by the nth expression. But that would mean that n is not Richardian. But if n is not Richardian, then that would define n as having the property described by the nth expression. And that property is Richardian. Hence there is a paradox.


But again, as in the case of the assumption of a list of English expressions for real numbers, the definition of the list itself cannot be in the same language as the list, and hence there cannot be an expression in the list that can refer to the definition of the list. Hence the supposed Richardian expression cannot occur in the list, and the supposed paradox disappears.


Other paradoxes


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