Richard's Paradox: a contradiction occurring in natural language descriptions of numbers; it is similar, though not identical, to the ►Berry paradox.

Jules Richard was a high school teacher in Dijon, France, who spent his free time reading up on infinity and number theory. In 1905 he sent a letter to the then-leading French science magazine:

I define a set of numbers S as follows: we compile a list of all possible combinations of two letters in alphabetic order, after that we list all possible combinations of three letters also in alphabetic order, then all possible combinations of four letters, and so forth. The same letter may occur multiple times in this list. For the sake of clarity we also include punctuation marks and spaces. Thus, for each natural number p we can find all possible combinations of p letters, punctuation marks, and spaces in the thus-compiled list. Since all that can be written in a finite number of words consists in an arrangement of letters, the list will contain everything that can be written.*

Since numbers are described by means of words and words consist of letters, some of the entries on the list will be definitions of numbers. Now let us delete all those entries that are not definitions of real numbers. Let u1 be the number defined by the first entry on the list, u2 the one defined by the second entry, u3 the one defined by the third entry, and so forth.** By this we have arranged all real numbers that can be defined by means of a finite number of words in an unambiguous order. Therefore, all those numbers constitute a countably infinite set. This is our set of numbers S. But here is the paradox. We can generate a real number that does not belong to the set S:

"This number N is obtained as follows. Let the digit immediately before the decimal point be 0 and the nth digit after the decimal point p+1, if p does not equal 8 or 9, and otherwise 1. Here, p is the nth digit after the decimal point of the nth number in set S."***

The number N does not belong to the set S, because if it were identical to any nth number of set S then its nth digit after the decimal point would have to coincide with the nth digit after the decimal point of that number, which according to the above construction is not the case. On the other hand, the number N is defined by a finite number of words, namely those that are above in the quotation marks. Therefore, according to its definition it would have to belong to the set S.

Richard also offered a solution to this paradox: The above description of the number N in quotation marks does not actually constitute a complete definition. Therefore, the number cannot occur anywhere on the list in the first place, for it requires for its complete definition the entire set S, and the latter is not fully defined until the list is complete. Thus, we have a circle, a so-called recursion in our definition. To avoid this circle, we have to stipulate that the set S may not be used in the definition of one of its elements.

However, Richard did not have any better luck than Olbers in this respect: his paradox quickly became famous, but his proposed solution triggered objections. The mathematician Peano considered the definition of the number N to be perfectly legitimate, even though it contains a recursion. According to Peano, the problem lies rather in the imprecise nature of natural language. Every definition uses a certain vocabulary, which in turn requires a metalanguage for its definition. In this sense natural languages are never without a circle — which is the cause of all of their paradoxes.

* The ►library of Babel mentioned in another article constitutes only a small part of this list, namely the one with p = 1312000, although in an unordered arrangement.

** In German, for example, u1 = 11, u2 = 8, u3 = 3. We leave the proof to the reader.

*** The number N begins also in the German list with 0.111...

© Johann Christian Lotter   ■  Infinity  ■  Links  ■  Forum