in Frege is the following: In other words, given the function is the variable and is the invariant part. So why not substitute the value for itself? Russell promptly wrote Frege a letter pointing out that: Frege promptly wrote back to Russell acknowledging the problem: While the problem had adverse personal consequences for both men (both had works at the printers that had to be emended), van Heijenoort observes that "The paradox shook the logicians' world, and the rumbles are still felt today. ... Russell's paradox, which uses the bare notions of set and element, falls squarely in the field of logic. The paradox was first published by Russell in The principles of mathematics (1903) and is discussed there in great detail ...". Russell, after six years of false starts, would eventually answer the matter with his 1908 theory of types by "propounding his axiom of reducibility. It says that any function is coextensive with what he calls a predicative function: a function in which the types of apparent variables run no higher than the types of the arguments". But this "axiom" was met with resistance from all quarters. The rejection of impredicatively defined mathematical objects (while accepting the natural numbers as classically understood) leads to the position in the philosophy of mathematics known as predicativism, advocated by Henri Poincaré and Hermann Weyl in his Das Kontinuum. Poincaré and Weyl argued that impredicative definitions are problematic only when one or more underlying sets are infinite. Ernst Zermelo in his 1908 "A new proof of the possibility of a well-ordering" presents an entire section "b. Objection concerning nonpredicative definition" where he argued against "Poincaré (1906, p. 307) [who states that] a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it". He gives two examples of impredicative definitions – (i) the notion of Dedekind chains and (ii) "in analysis wherever the maximum or minimum of a previously defined "completed" set of numbers is used for further inferences. This happens, for example, in the well-known Cauchy proof...". He ends his section with the following observation: "A definition may very well rely upon notions that are equivalent to the one being defined; indeed, in every definition definiens and definiendum are equivalent notions, and the strict observance of Poincaré's demand would make every definition,