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Single Idea 14436

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts ]

Full Idea

There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'.

Gist of Idea

A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum

Source

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

Book Ref

Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.69

Related Idea

Idea 14437 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]

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]