To escape this outcome, Russell postulated his " axiom of reducibility ".
2.
But the axiom of reducibility proposes that " in theory " a reduction " all the way down " is possible.
3.
They also had to make several compromises in order to develop so much of mathematics, such as an " axiom of reducibility ".
4.
Holmes has shown that NFP has the same consistency strength as the predicative theory of types of " Principia Mathematica " without the Axiom of reducibility.
5.
Russell, after six years of false starts, would eventually answer the matter with his 1908 theory of types by " propounding his " axiom of reducibility ".
6.
"' Russell abandons the axiom of reducibility "': In the second edition of " Principia Mathematica " ( 1927 ) he acknowledges Wittgenstein's argument.
7.
Russell and Whitehead found it impossible to develop mathematics while maintaining the difference between predicative and non-predicative functions, so introduced the axiom of reducibility, saying that for every non-predicative function there is a predicative function taking the same values.
8.
However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility.
9.
But because the stipulations of the ramified theory would prove ( to quote Quine ) " onerous ", Russell in his 1908 " Mathematical logic as based on the theory of types " also would propose his " axiom of reducibility ".
10.
Now he fully embraces the matrix notion and declares " A " function can only appear in a matrix through its values " " ( but demurs in a footnote : " It takes the place ( not quite adequately ) of the axiom of reducibility " ).
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