Editing Logicism (section)

===Gödel's criticism and suggestions===
Gödel, in his 1944 work, identifies the place where he considers Russell's logicism to fail and offers suggestions to rectify the problems. He submits the "vicious circle principle" to re-examination, splitting it into three parts "definable only in terms of", "involving" and "presupposing". It is the first part that "makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of mathematics itself". Since, he argues, mathematics sees to rely on its inherent impredicativities (e.g. "real numbers defined by reference to all real numbers"), he concludes that what he has offered is "a proof that the vicious circle principle is false [rather] than that classical mathematics is false" (all quotes Gödel 1944:127).

'''Russell's no-class theory is the root of the problem''': Gödel believes that impredicativity is not "absurd", as it appears throughout mathematics. Russell's problem derives from his "constructivistic (or nominalistic"<ref>Perry observes that Plato and Russell are "enthusiastic" about "universals", then in the next sentence writes: " 'Nominalists' think that all that particulars really have in common are the words we apply to them" (Perry in his 1997 Introduction to Russell 1912:xi). Perry adds that while your sweatshirt and mine are different objects generalized by the word "sweatshirt", you have a relation to yours and I have a relation to mine. And Russell "treated relations on par with other universals" (p. xii). But Gödel is saying that Russell's "no-class" theory denies the numbers the status of "universals".</ref>) standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . a notion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction" (p.&nbsp;128).

Indeed, Russell's "no class" theory, Gödel concludes:
:"is of great interest as one of the few examples, carried out in detail, of the tendency to eliminate assumptions about the existence of objects outside the "data" and to replace them by constructions on the basis of these data<sup>33</sup>. The "data" are to understand in a relative sense here; i.e. in our case as logic without the assumption of the existence of classes and concepts]. The result has been in this case essentially negative; i.e. the classes and concepts introduced in this way do not have all the properties required from their use in mathematics. . . . All this is only a verification of the view defended above that logic and mathematics (just as physics) are built up on axioms with a real content which cannot be explained away" (p. 132)

He concludes his essay with the following suggestions and observations:
:"One should take a more conservative course, such as would consist in trying to make the meaning of terms "class" and "concept" clearer, and to set up a consistent theory of classes and concepts as objectively existing entities.  This is the course which the actual development of mathematical logic has been taking and which Russell himself has been forced to enter upon in the more constructive parts of his work. Major among the attempts in this direction . . . are the simple [[type theory|theory of types]] . . . and [[axiomatic set theory]], both of which have been successful at least to this extent, that they permit the derivation of modern mathematics and at the same time avoid all known paradoxes . . . ¶ It seems reasonable to suspect that it is this incomplete understanding of the foundations which is responsible for the fact that mathematical logic has up to now remained so far behind the high expectations of Peano and others . . .." (p. 140)