Axiom of Reducibility - History

History

With Russell's discovery (1901, 1902) of a paradox in Gottlob Frege's 1879 Begriffsschrift and Frege's acknowledgment of the same (1902), Russell tentatively introduced his solution as "Appendix B: Doctrine of Types" in his 1903 Principles of Mathematics. This contradiction can be stated as "the class of all classes that do not contain themselves as elements". At the end of this appendix Russell asserts that his "doctrine" would solve the immediate problem posed by Frege, but "...there is at least one closely analogous contradiction which is probably not soluble by this doctrine. The totality of all logical objects, or of all propositions, involves, it would seem a fundamental logical difficulty. What the complete solution of the difficulty may be, I have not succeeded in discovering; but as it affects the very foundations of reasoning..."

By the time of his 1908 Mathematical logic as based on the theory of types Russell had studied "the contradictions" (among them the Epimenides paradox, the Burali-Forti paradox, and Richard's paradox) and concluded that "In all the contradictions there is a common characteristic, which we may describe as self-reference or reflexiveness".

In 1903, Russell defined predicative functions as those whose order is one more than the highest order function occurring in the expression of the function. While these were fine for the situation, impredicative functions had to be disallowed:

"A function whose argument is an individual and whose value is always a first-order proposition will be called a first-order function. A function involving a first-order function or proposition as apparent variable will be called a second-order function, and so on. A function of one variable which is of the order next above that of its argument will be called a predicative function; the same name will be given to a function of several variables . . .."

He repeats this definition in a slightly different way later in the paper (together with a subtle prohibition that they would express more clearly in 1913): "A predicative function of x is one whose values are propositions of the type next above that of x, if x is an individual or a proposition, or that of values of x if x is a function. It may be described as one in which the apparent variables, if any, are all of the same type as x or of lower type; and a variable is of lower type than x if it can significantly occur as argument to x, or as argument to an argument to x, and so forth."

This usage carries over to Alfred North Whitehead and Russell's 1913 Principia Mathematica wherein the authors devote an entire subsection of their Chapter II: "The Theory of Logical Types" to subchapter I. The Vicious-Circle Principle: "We will define a function of one variable as predicative when it is of the next order above that of its argument, i.e. of the lowest order compatible with its having that argument. . . A function of several arguments is predicative if there is one of its arguments such that, when the other arguments have values assigned to them, we obtain a predicative function of the one undetermined argument."

They again propose the definition of a predicative function as one that does not violate The Theory of Logical Types. Indeed the authors assert such violations are "incapable " and "impossible":

"We are thus lead to the conclusion, both from the vicious-circle principle and from direct inspection, that the functions to which a given object a can be an argument are incapable of being arguments to each other, and that they have no term in common with the functions to which they can be arguments. We are thus led to construct a hierarchy."

The authors stress the word impossible:

". . .if we are not mistaken, that not only is it impossible for a function φz^ to have itself or anything derived from it as argument, but that, if ψz^ is another function such there are arguments a with which both "φa" and "ψa" are significant, then ψz^ and anything derived from it cannot significantly be argument to φz^."