hence all of science, impossible". Zermelo's example of minimum and maximum of a previously defined "completed" set of numbers reappears in Kleene 1952:42-42 where Kleene uses the example of least upper bound in his discussion of impredicative definitions; Kleene does not resolve this problem. In the next paragraphs he discusses Weyl's attempt in his 1918 Das Kontinuum (The Continuum) to eliminate impredicative definitions and his failure to retain the "theorem that an arbitrary non-empty set of real numbers having an upper bound has a least upper bound (cf. also Weyl 1919)". Ramsey argued that "impredicative" definitions can be harmless: for instance, the definition of "tallest person in the room" is impredicative, since it depends on a set of things of which it is an element, namely the set of all persons in the room. Concerning mathematics, an example of an impredicative definition is the smallest number in a set, which is formally defined as: if and only if for all elements of , is less than or equal to , and is in . Burgess (2005) discusses predicative and impredicative theories at some length, in the context of Frege's logic, Peano arithmetic, second-order arithmetic, and axiomatic set theory. See also Gödel, Escher, Bach Impredicative polymorphism Logicism Richard's paradox Notes References PlanetMath article on predicativism John Burgess, 2005. Fixing Frege. Princeton Univ. Press. Solomon Feferman, 2005, "Predicativity" in The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press: 590–624. Stephen C. Kleene 1952 (1971 edition), Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY, . In particular cf. his §11 The Paradoxes (pp. 36–40) and §12 First inferences from the paradoxes IMPREDICATIVE DEFINITION (p. 42). He states that his 6 or so (famous) examples of paradoxes (antinomies) are all examples of impredicative definition, and says that Poincaré (1905–6, 1908) and Russell (1906, 1910) "enunciated the cause of the paradoxes to lie in these impredicative definitions" (p. 42), however, "parts of mathematics we want to retain, particularly analysis, also contain impredicative definitions." (ibid). Weyl in his 1918 ("Das Kontinuum") attempted to derive as much of analysis as was possible without the use of impredicative definitions, "but not the theorem that an arbitrary non-empty set M of real numbers having an upper bound has a least upper bound (CF. also Weyl 1919)" (p. 43). Hans Reichenbach 1947, Elements of Symbolic Logic, Dover Publications, Inc., NY, . Cf. his §40. The antinomies and