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Single Idea 14459

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility ]

Full Idea

The Axiom of Reducibility says 'There is a type of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question'. ..It involves all that is really essential in the theory of classes. But is it true?

Clarification

'a-functions' are all the functions which can take object a as an argument

Gist of Idea

Reducibility: a family of functions is equivalent to a single type of function

Source

Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII)

Book Ref

Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.191

A Reaction

I take this to say that in the theory of types, it is possible to reduce each level of type down to one type.

The 54 ideas from 'Introduction to Mathematical Philosophy'

8468 The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false [Russell, by Orenstein]
8469 Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein]
13414 For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf]
14420 Infinity and continuity used to be philosophy, but are now mathematics [Russell]
8745 Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell]
14423 '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
14424 Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell]
14421 Discovering that 1 is a number was difficult [Russell]
14422 Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
14425 A number is something which characterises collections of the same size [Russell]
14430 If a relation is symmetrical and transitive, it has to be reflexive [Russell]
14426 A definition by 'extension' enumerates items, and one by 'intension' gives a defining property [Russell]
14427 We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell]
14428 Members define a unique class, whereas defining characteristics are numerous [Russell]
14429 Classes are logical fictions, made from defining characteristics [Russell]
14431 The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
14441 The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell]
10450 Russell admitted that even names could also be used as descriptions [Russell, by Bach]
13510 Could a number just be something which occurs in a progression? [Russell, by Hart,WD]
14432 'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a [Russell]
14435 The essence of individuality is beyond description, and hence irrelevant to science [Russell]
14434 What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell]
14433 Mathematically expressed propositions are true of the world, but how to interpret them? [Russell]
14436 A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
14439 A complex number is simply an ordered couple of real numbers [Russell]
14438 New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell]
14440 We may assume that there are infinite collections, as there is no logical reason against them [Russell]
14442 If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell]
14443 The British parliament has one representative selected from each constituency [Russell]
14445 Choice shows that if any two cardinals are not equal, one must be the greater [Russell]
14444 Choice is equivalent to the proposition that every class is well-ordered [Russell]
14446 We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell]
14447 Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell]
14449 There is always something psychological about inference [Russell]
12197 Inferring q from p only needs p to be true, and 'not-p or q' to be true [Russell]
14450 All forms of implication are expressible as truth-functions [Russell]
14451 Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts [Russell]
14454 An argument 'satisfies' a function φx if φa is true [Russell]
14453 The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M? [Russell]
14452 All the propositions of logic are completely general [Russell]
14456 'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity [Russell]
12444 Logic is concerned with the real world just as truly as zoology [Russell]
14458 Asking 'Did Homer exist?' is employing an abbreviated description [Russell]
14457 Names are really descriptions, except for a few words like 'this' and 'that' [Russell]
7311 The only genuine proper names are 'this' and 'that' [Russell]
14455 'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not [Russell]
14459 Reducibility: a family of functions is equivalent to a single type of function [Russell]
14461 Propositions about classes can be reduced to propositions about their defining functions [Russell]
14460 If something is true in all possible worlds then it is logically necessary [Russell]
14465 Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell]
14462 In modern times, logic has become mathematical, and mathematics has become logical [Russell]
14464 Logic can be known a priori, without study of the actual world [Russell]
10057 Logic can only assert hypothetical existence [Russell]
14463 Existence can only be asserted of something described, not of something named [Russell]