Richard's paradox

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In logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics. The paradox was also a motivation in the development of predicative mathematics.

Description[edit]

The original statement of the paradox, due to Richard (1905), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers.

The paradox begins with the observation that certain expressions in English unambiguously define real numbers, while other expressions in English do not. For example, "The real number whose integer part is 17 and whose nth decimal place is 0 if n is even and 1 if n is odd" defines the real number 17.1010101..., while the phrase "the capital of England" does not define a real number.

Thus there is an infinite list of English phrases (where each phrase is of finite length, but lengths vary in the list) that unambiguously define real numbers; arrange this list by length and then order lexicographically, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1.

The preceding two paragraphs are an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers rn. However, r was constructed so that it cannot equal any of the rn. This is the paradoxical contradiction.

Analysis and relationship with metamathematics[edit]

Richard's paradox leaves an untenable contradiction, which must be analyzed to find an error.

The proposed definition of the new real number r clearly contains a finite string of characters, and hence it appears at first to be a definition of a real number. However, the definition refers to definability-in-English itself. If it were possible to determine which English expressions actually do define a real number, and which do not, then the paradox would go through. Thus the resolution of Richard's paradox is that there is no way to unambiguously determine exactly which English sentences are definitions of real numbers (see Good 1966). That is, there is no way to describe in a finite number of words how to tell whether an arbitrary English expression is a definition of a real number. This is not surprising, as the ability to make this determination would also imply the ability to solve the halting problem and perform any other non-algorithmic calculation that can be described in English.

A similar phenomenon occurs in formalized theories that are able to refer to their own syntax, such as Zermelo–Fraenkel set theory (ZFC). Say that a formula φ(x) defines a real number if there is exactly one real number r such that φ(r) holds. Then it is not possible to define, in ZFC, the set of all (Gödel numbers of) formulas that define real numbers. For, if it were possible to define this set, it would be possible to diagonalize over it to produce a new definition of a real number, following the outline of Richard's paradox above. Note that the set of formulas that define real numbers may exist, as a set F; the limitation of ZFC is that there is no formula that defines F without reference to other sets. This is closely related to Tarski's indefinability theorem.

The example of ZFC illustrates the importance of distinguishing the metamathematics of a formal system from the statements of the formal system itself. The property D(φ) that a formula φ of ZFC defines a unique real number is not itself expressible in ZFC, but must be studied in the metatheory used to formalize ZFC. From this viewpoint, Richard's paradox results from treating a construction in the metatheory (the enumeration of all statements in the original system that define real numbers) as if that construction could be conducted in the original system.

Variation: Richardian numbers[edit]

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 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 natural number 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 natural 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. 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: "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.

Relation to predicativism[edit]

Another viewpoint on Richard's paradox relates to mathematical predicativism. In this view, the real numbers are defined in stages, with each stage only making reference to previous stages and other things that have already been defined. From a predicative viewpoint it is not valid to quantify over all real numbers in the process of generating a new real number, because this is believed to lead to a vicious-circle problem in the definitions. Set theories such as ZFC are not based on this sort of predicative framework, and allow impredicative definitions.

Richard (1905) presented a solution to the paradox from the viewpoint of predicativisim. Richard claimed that the flaw in the paradoxical construction was that the expression for the construction of the real number r does not actually unambiguously define a real number, because the statement refers to the construction of an infinite set of real numbers, of which r itself is a part. Thus, Richard says, the real number r will not be included as any rn, because the definition of r does not meet the criteria for being included in the sequence of definitions used to construct the sequence rn. Contemporary mathematicians agree that the definition of r is invalid, but for a different reason. They believe the definition of r is invalid because there is no well-defined notion of when an English phrase defines a real number, and so there is no unambiguous way to construct the sequence rn.

Although Richard's solution to the paradox did not gain favor with mathematicians, predicativism is an important part of the study of the foundations of mathematics. Predicativism was first studied in detail by Henri Poincaré and Hermann Weyl in Das Kontinuum, where they showed that much of elementary real analysis can be conducted in a predicative manner starting with only the natural numbers. More recently, predicativism has been studied by Solomon Feferman, who has used proof theory to explore the relationship between predicative and impredicative systems.

See also[edit]

References[edit]

  • Fraenkel, Abraham; Bar-Hillel, Yehoshua & Levy, Azriel (1973). Foundations of Set Theory. With the collaboration of Dirk van Dalen (Second ed.). Amsterdam: Noord-Hollandsche. ISBN 0-7204-2270-1. 
  • Good, I. J. (1966). "A Note on Richard's Paradox". Mind 75 (299): 431. doi:10.1093/mind/LXXV.299.431. 
  • Richard, Jules (1905). Les Principes des Mathématiques et le Problème des Ensembles. Revue Générale des Sciences Pures et Appliquées.  Translated in Heijenoort, J. van, ed. (1964). Source Book in Mathematical Logic 1879-1931. Cambridge, MA: Harvard University Press. 

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