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1908 3 2 wiener 1914 3 3 wittgenstein 1918 3 4 russell 1919 3 5 skolem 1922 3 6 russell 1927 3 7 von neumann 1925 3 8 david hilbert 1927 3 9 ramsey 1925 3 10 gödel 1944 3 11 quine 1967 3 12 kleene 1952 4 notes 5 references 6 external links toggle the table of contents axiom of reducibility add languages add links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia axiom in russell s ramified theory of types 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 1 history edit with russell s discovery 1901 1902 2 of a paradox in gottlob frege s 1893 grundgesetze der arithmetik and frege s acknowledgment of the same 1902 russell tentatively introduced his solution as appendix b doctrine of types in his 1903 the principles of mathematics 3 this contradiction can be stated as the class of all classes that do not contain themselves as elements 4 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 5 by the time of his 1908 mathematical logic as based on the theory of types 6 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 7 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 etc 8 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 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 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 10 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 to achieve and impossible we are thus led 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 11 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 12 russell s axiom of reducibility edit the axiom of reducibility states that any truth function i e propositional function can be expressed by a formally equivalent predicative truth function 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 13 in his words thus a predicative function of an individual is a first order function and for higher types of arguments predicative functions take the place that first order functions take in respect of individuals we assume then that every function is equivalent for all its values to some predicative function of the same argument this assumption seems to be the essence of the usual assumption of classes modern sets we will call this assumption the axiom of classes or the axiom of reducibility 14 for relations functions of two variables such as for all x and for all y those values for which f x y is true i e x y f x y russell assumed an axiom of relations or the same axiom of reducibility 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 a m i e the particular a j is a constant or a parameter held constant then evaluate f a m y n across all the n instances of possible y n for all y n evaluate f a 1 y n then for all y n evaluate f a 2 y n etc until all the x a m are exhausted this would create an m by n matrix of values true or unknown in this exposition the use of indices is a modern convenience in 1908 russell made no mention of this matrix of x y values that render a two place function e g relation true but by 1913 he has introduced a matrix like concept into function in 12 of principia mathematica 1913 he defines a matrix as any function of however many variables which does not involve any apparent variables then any possible function other than a matrix is derived from a matrix by means of generalisation i e by considering the proposition which asserts that the function in question is true with all possible values or with some values of one of the arguments the other argument or arguments remaining undetermined 15 for example if one asserts that y f x y is true then x is the apparent variable because it is unspecified russell now defines a matrix of individuals as a first order matrix and he follows a similar process to define a second order matrix etc finally he introduces the definition of a predicative function a function is said to be predicative when it is a matrix 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 1913 127 meaning the function contains no apparent variables matrix or predicative function is a primitive idea 16 from this reasoning he then uses the same wording to propose the same axioms of reducibility as he did in his 1908 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 an intuitive version of this notion appeared in frege s 1879 begriffsschrift translated in van heijenoort 1967 23 russell s 1903 followed closely the work of frege cf russell 1903 505ff russell worried that it is necessary to give sense to the couple to distinguish the referent from the relatum thus a couple becomes essentially distinct from a class of two terms and must itself be introduced as a primitive idea it would seem viewing the idea philosophically that sense can only be derived from some relational proposition it seems therefore more correct to take an intensional view of relations and to identify them rather with class concepts than with classes 18 as shown below norbert wiener 1914 reduced the notion of relation to class by his definition of an ordered pair criticism edit zermelo 1908 edit 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 19 zermelo countered a definition may very well rely upon notions that are equivalent to the one being defined indeed in every definition definiens and definiendum are equivalent notions and the strict observance of poincaré s demand would make every definition hence all of science impossible 20 wiener 1914 edit in his 1914 a simplification of the logic of relations norbert wiener removed the need for the axiom of reducibility as applied to relations between two variables x and y e g φ x y he did this by introducing a way to express a relation as a set of ordered pairs it will be seen that what we have done is practically to revert to schröder s treatment of a relation as a class set of ordered couples 21 van heijenoort observes that b y giving a definition of the ordered pair of two elements in terms of class operations the note reduced the theory of relations to that of classes 22 but wiener opined that while he had dispatched russell and whitehead s two variable version of the axiom 12 11 the single variable version of the axiom of reducibility for axiom 12 1 in principia mathematica was still necessary 23 wittgenstein 1918 edit ludwig wittgenstein while imprisoned in a prison camp finished his tractatus logico philosophicus his introduction credits the great works of frege and the writings of my friend bertrand russell not a self effacing intellectual he pronounced that the truth of the thoughts communicated here seems to me unassailable and definitive i am therefore of the opinion that the problems have in essentials been finally solved 24 so given such an attitude it is no surprise that russell s theory of types comes under criticism 3 33 in logical syntax the meaning of a sign ought never to play a role it must admit of being established without mention being thereby made of the meaning of a sign it ought to presuppose only the description of the expressions 3 331 from this observation we get a further view into russell s theory of types russell s error is shown by the fact that in drawing up his symbolic rules he has to speak of the meaning of the signs 3 332 no proposition can say anything about itself because the proposition sign cannot be contained in itself that is the whole theory of types 3 333 a function cannot be its own argument because the functional sign already contains the prototype of its own argument and it cannot contain itself herewith russell s paradox vanishes 25 this appears to support the same argument russell uses to erase his paradox 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 it is clear that the laws of logic cannot themselves obey further logical laws there is not as russell supposed for every type a special law of contradiction but one is sufficient since it is not applied to itself 6 1231 the mark of logical propositions is not their general validity to be general is only to be accidentally valid for all things an ungeneralised proposition can be tautologous just as well as a generalised one 6 1232 logical general validity we could call essential as opposed to accidental general validity e g of the proposition all men are mortal propositions like russell s axiom of reducibility are not logical propositions and this explains our feeling that if true they can only be true by a happy chance 6 1233 we can imagine a world in which the axiom of reducibility is not valid but it is clear that logic has nothing to do with the question of whether our world is really of this kind or not 26 russell 1919 edit 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 he concludes that we cannot accept class as a primitive idea the symbols for classes are mere conveniences and classes are logical fictions or as we say incomplete symbols classes cannot be regarded as part of the ultimate furniture of the world p 146 the reason for this is because of the problem of impredicativity classes cannot be regarded as a species of individuals on account of the contradiction about classes which are not members of themselves and because we can prove that the number of classes is greater than the number of individuals etc what he then does is propose 5 obligations that must be satisfied with respect to a theory of classes and the result is his axiom of reducibility he states that this axiom is a generalised form of leibniz s identity of indiscernibles p 155 but he concludes leibniz s assumption is not necessarily true for all possible predicates in all possible worlds so he concludes that i do not see any reason to believe that the axiom of reducibility is logically necessary which is what would be meant by saying that it is true in all possible worlds the admission of this axiom ...