To escape this outcome, Russell postulated his " axiom of reducibility ", which asserts that to any property belonging to an order above the lowest, there is a coextensive property ( i . e . one possessed by exactly the same objects ) of order 0.
12.
Observe that while Wiener " reduced " the relational * 12.11 form of the axiom of reducibility he " did not " reduce nor otherwise change the propositional-function form * 12.1; indeed he declared this " essential to the treatment of identity, descriptions, classes and relations ".
13.
Thus Russell has virtually abandoned the axiom of reducibility, but in his last paragraphs he states that from " our present primitive propositions " he cannot derive " Dedekindian relations and well-ordered relations " and observes that if there is a new axiom to replace the axiom of reducibility " it remains to be discovered ".
14.
Thus Russell has virtually abandoned the axiom of reducibility, but in his last paragraphs he states that from " our present primitive propositions " he cannot derive " Dedekindian relations and well-ordered relations " and observes that if there is a new axiom to replace the axiom of reducibility " it remains to be discovered ".
15.
As Stewart Shapiro explains in his " Thinking About Mathematics ", Russell's attempts to solve the paradoxes led to the ramified theory of types, which, though it is highly complex and relies on the doubtful axiom of reducibility, actually manages to solve both syntactic and semantic paradoxes at the expense of rendering the logicist project suspect and introducing much complexity in the PM system.
16.
Russell arrived at this disheartening conclusion : that " the theory of ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealt with . . . . Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom . " ( " PM " 1927 : xiv ).
17.
Now equipped with the matrix notion, " PM " can assert its controversial axiom of reducibility : a function of one or two variables ( two being sufficient for " PM "'s use ) " where all its values are given " ( i . e ., in its matrix ) is ( logically ) equivalent ( " a " " ) to some " predicative " function of the same variables.
18.
But, Kleene wonders, " on what grounds should we believe in the axiom of reducibility ? " He observes that, whereas " Principia Mathematica " is presented as derived from " intuitively "-derived axioms that " were intended to be believed about the world, or at least to be accepted as plausible hypotheses concerning the world [, ] . . . if properties are to be constructed, the matter should be settled on the basis of constructions, not by an axiom . " Indeed, he quotes Whitehead and Russell ( 1927 ) questioning their own axiom : " clearly it is not the sort of axiom with which we can rest content ".
19.
Hilbert states that, with regard to this system, i . e . " Russell and Whitehead's theory of foundations [, ] . . . the foundation that it provides for mathematics rests, first, upon the axiom of infinity and, then upon what is called the axiom of reducibility, and both of these axioms are genuine contentual assumptions that are not supported by a consistency proof; they are assumptions whose validity in fact remains dubious and that, in any case, my theory does not require . . . reducibility is not presupposed in my theory . . . the execution of the reduction would be required only in case a proof of a contradiction were given, and then, according to my proof theory, this reduction would always be bound to succeed ."
