The axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types.
Russell devised and introduced the axiom in an attempt to manage the contradictions he had discovered in his analysis of set theory.
[emphasis added][9]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.
It made its first appearance in Bertrand Russell's (1908) Mathematical logic as based on the theory of types, but only after some five years of trial and error.
In 1903, he proposed a possible process of evaluating such a 2-place function by comparing the process to double integration: One after another, plug into x definite values am (i.e. the particular aj is "a constant" or a parameter held constant), then evaluate f(am,yn) across all the n instances of possible yn.
It will be observed that, in a hierarchy in which all the variables are individuals or matrices, a matrix is the same thing as an elementary function [cf.
As an aside, Russell in his 1903 considered, and then rejected, "a temptation to regard a relation as definable in extension as a class of couples",[17] i.e. the modern set-theoretic notion of ordered pair.
[18] As shown below, Norbert Wiener (1914) reduced the notion of relation to class by his definition of an ordered pair.
The outright prohibition implied by Russell's axiom of reducibility was roundly criticised by Ernst Zermelo in his 1908 Investigations in the foundations of set theory I, stung as he was by a demand similar to that of Russell that came from Poincaré: According to Poincaré (1906, p. 307) 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.
This "using the signs" to "speak of the signs" Russell criticises in his introduction that preceded the original English translation: What causes hesitation is the fact that, after all, Mr Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exit.This problem appears later when Wittgenstein arrives at this gentle disavowal of the axiom of reducibility—one interpretation of the following is that Wittgenstein is saying that Russell has made (what is known today as) a category error; Russell has asserted (inserted into the theory) a "further law of logic" when all the laws (e.g. the unbounded Sheffer stroke adopted by Wittgenstein) have already been asserted: 6.123 6.1231 6.1232 6.1233 Bertrand Russell in his 1919 Introduction to Mathematical Philosophy, a non-mathematical companion to his first edition of PM, discusses his Axiom of Reducibility in Chapter 17 Classes (pp. 146ff).
(p. 155)Thoralf Skolem in his 1922 Some remarks on axiomatised set theory took a less than positive attitude toward "Russell and Whitehead" (i.e. their work Principia Mathematica): Until now, so far as I know, only one such system of axioms has found rather general acceptance, namely that constructed by Zermelo (1908).
Russell and Whitehead, too, constructed a system of logic that provides a foundation for set theory; if I am not mistaken, however, mathematicians have taken but little interest in it.
... [But the consequences of his logic are that] the theory of infinite Dedekindian and well-ordering collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with.
5.5422 A possible interpretation of Wittgenstein's stance is that the thinker A i.e. 'p' is identically the thought p, in this way the "soul" remains a unit and not a composite.
John von Neumann in his 1925 "An axiomatisation of set theory" wrestled with the same issues as did Russell, Zermelo, Skolem, and Fraenkel.
They arrived at entirely different results [from the set theorists], but the over-all effect of their activity seems to me outright devastating.
[34] While he mentions the efforts of David Hilbert to prove the consistency of his axiomatisation of mathematics[35] von Neumann placed him in the same group as Russell.
Rather, von Neumann considered his proposal to be "in the spirit of the second group ... We must, however, avoid forming sets by collecting or separating elements [durch Zusammenfassung oder Aussonderung von Elementen], and so on, as well as eschew the unclear principle of 'definiteness' that can still be found in Zermelo.
"[37] David Hilbert's axiomatic system that he presents in his 1925 The Foundations of Mathematics is the mature expression of a task he set about in the early 1900s but let lapse for a while (cf.
He observes that the first edition of PM "abandoned" the realist (constructivistic) "attitude" with his proposal of the axiom of reducibility (p. 133).
However, within the introduction to the second edition of PM (1927) Gödel asserts "the constructivistic attitude is resumed again" (p. 133) when Russell "dropped" of the axiom of reducibility in favour of the matrix (truth-functional) theory; Russell "stated explicitly that all primitive predicates belong to the lowest type and that the only purpose of variables (and evidently also of constants) is to make it possible to assert more complicated truth-functions of atomic propositions ... [i.e.] the higher types and orders are solely a façon de parler" (p. 134).
However, from such finite strings the whole of mathematics cannot be constructed because they cannot be "analyzed", i.e. reducible to the law of identity or disprovable by a negations of the law: Even the theory of integers is non-analytic, provided that one requires of the rules of elimination that they allow one actually to carry out the elimination in a finite number of steps in each case.44 (44Because this would imply the existence of a decision procedure for all arithmetical propositions.
(p. 139)But he observes that "this procedure seems to presuppose arithmetic in some form or other" (p. 134), and he states in the next paragraph that "the question of whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy must be considered as unsolved."
This is the course which the actual development of mathematical logic has been taking ... Major among the attempts in this direction ... are the simple 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.
(p. 140)In a critique that also discusses the pros and cons of Ramsey (1931)[44] W. V. O. Quine calls Russell's formulation of "types" to be "troublesome ... the confusion persists as he attempts to define 'nth order propositions'... the method is indeed oddly devious ... the axiom of reducibility is self-effacing", etc.
[45] Like Stephen Kleene, Quine observes that Ramsey (1926)[46] divided the various paradoxes into two varieties (i) "those of pure set theory" and (ii) those derived from "semantic concepts such as falsity and specifiability", and Ramsey believed that the second variety should have been left out of Russell's solution.
Quine ends with the opinion that "because of the confusion of propositions with sentences, and of attributes with their expressions, Russell's purported solution of the semantic paradoxes was enigmatic anyway.
First inferences from the paradoxes" (subchapter "LOGICISM"), Stephen Kleene (1952) traces the development of Russell's theory of types: To adapt the logicistic [sic] construction of mathematics to the situation arising from the discovery of the paradoxes, Russell excluded impredicative definitions by his ramified theory of types (1908, 1910).
"[54] Kleene ends this discussion with quotes from Weyl (1946) that "the system of Principia Mathematica ... [is founded on] a sort of logician's paradise" and anyone "who is ready to believe in this 'transcendental world' could also accept the system of axiomatic set theory (Zermelo, Fraenkel, etc), which, for the deduction of mathematics, has the advantage of being simpler in structure.