Relation To Predicativism
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.
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