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Single Idea 14141

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers ]

Full Idea

The ordinal numbers are defined by some relation to mathematical induction.

Gist of Idea

Ordinals are defined through mathematical induction

Source

Bertrand Russell (The Principles of Mathematics [1903], §290)

Book Ref

Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.312

The 92 ideas from 'The Principles of Mathematics'

15894 Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine]
18722 Negations are not just reversals of truth-value, since that can happen without negation [Wittgenstein on Russell]
15895 Russell discovered the paradox suggested by Burali-Forti's work [Russell, by Lavine]
18254 Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett]
7530 Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
11010 Being is what belongs to every possible object of thought [Russell]
11849 It at least makes sense to say two objects have all their properties in common [Wittgenstein on Russell]
19164 If propositions are facts, then false and true propositions are indistinguishable [Davidson on Russell]
14102 What is true or false is not mental, and is best called 'propositions' [Russell]
14103 Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
14104 Constants are absolutely definite and unambiguous [Russell]
14105 There seem to be eight or nine logical constants [Russell]
14106 Implication cannot be defined [Russell]
14107 Terms are identical if they belong to all the same classes [Russell]
14108 It would be circular to use 'if' and 'then' to define material implication [Russell]
14109 The study of grammar is underestimated in philosophy [Russell]
7781 I call an object of thought a 'term'. This is a wide concept implying unity and existence. [Russell]
14110 Proposition contain entities indicated by words, rather than the words themselves [Russell]
14111 A proposition is a unity, and analysis destroys it [Russell]
14112 A set has some sort of unity, but not enough to be a 'whole' [Russell]
14113 The null class is a fiction [Russell]
14114 Variables don't stand alone, but exist as parts of propositional functions [Russell]
14115 Definition by analysis into constituents is useless, because it neglects the whole [Russell]
14116 Numbers were once defined on the basis of 1, but neglected infinities and + [Russell]
14117 Numbers are properties of classes [Russell]
14118 We can define one-to-one without mentioning unity [Russell]
14119 We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
14120 Counting explains none of the real problems about the foundations of arithmetic [Russell]
14121 The part-whole relation is ultimate and indefinable [Russell]
14122 Analysis gives us nothing but the truth - but never the whole truth [Russell]
14123 Some quantities can't be measured, and some non-quantities are measurable [Russell]
10583 Abstraction principles identify a common property, which is some third term with the right relation [Russell]
10582 The principle of Abstraction says a symmetrical, transitive relation analyses into an identity [Russell]
10584 A certain type of property occurs if and only if there is an equivalence relation [Russell]
14124 Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
14125 Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
14126 Order rests on 'between' and 'separation' [Russell]
14127 Order depends on transitive asymmetrical relations [Russell]
10586 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness [Russell]
10585 Symmetrical and transitive relations are formally like equality [Russell]
14128 Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell]
14129 Ordinals presuppose two relations, where cardinals only presuppose one [Russell]
9977 Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
14132 Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
14133 There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
14134 Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell]
14135 Real numbers are a class of rational numbers (and so not really numbers at all) [Russell]
14137 'Any' is better than 'all' where infinite classes are concerned [Russell]
14138 You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell]
14140 For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell]
14139 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell]
14142 Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell]
14141 Ordinals are defined through mathematical induction [Russell]
14143 ω names the whole series, or the generating relation of the series of ordinal numbers [Russell]
14144 Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell]
14145 For Cantor ordinals are types of order, not numbers [Russell]
14146 We aren't sure if one cardinal number is always bigger than another [Russell]
14147 Denying mathematical induction gave us the transfinite [Russell]
14149 The Achilles Paradox concerns the one-one correlation of infinite classes [Russell]
14148 Infinite regresses have propositions made of propositions etc, with the key term reappearing [Russell]
14151 Pure geometry is deductive, and neutral over what exists [Russell]
14152 In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
14154 Geometry throws no light on the nature of actual space [Russell]
14153 In geometry, empiricists aimed at premisses consistent with experience [Russell]
14155 Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG]
14156 Mathematicians don't distinguish between instants of time and points on a line [Russell]
14158 Quantity is not part of mathematics, where it is replaced by order [Russell]
14159 In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives [Russell]
14160 Space is the extension of 'point', and aggregates of points seem necessary for geometry [Russell]
14161 Many things have being (as topics of propositions), but may not have actual existence [Russell]
22303 It makes no sense to say that a true proposition could have been false [Russell]
14162 Mathematics doesn't care whether its entities exist [Russell]
14163 Four classes of terms: instants, points, terms at instants only, and terms at instants and points [Russell]
14166 Unities are only in propositions or concepts, and nothing that exists has unity [Russell]
14164 The only unities are simples, or wholes composed of parts [Russell]
14165 Analysis falsifies, if when the parts are broken down they are not equivalent to their sum [Russell]
14167 The only classes are things, predicates and relations [Russell]
14169 The 'universe' can mean what exists now, what always has or will exist [Russell]
14168 Occupying a place and change are prior to motion, so motion is just occupying places at continuous times [Russell]
14170 Change is obscured by substance, a thing's nature, subject-predicate form, and by essences [Russell]
14171 Force is supposed to cause acceleration, but acceleration is a mathematical fiction [Russell]
14172 Moments and points seem to imply other moments and points, but don't cause them [Russell]
14173 What exists has causal relations, but non-existent things may also have them [Russell]
14174 The laws of motion and gravitation are just parts of the definition of a kind of matter [Russell]
14175 We can drop 'cause', and just make inferences between facts [Russell]
14176 "The death of Caesar is true" is not the same proposition as "Caesar died" [Russell]
21555 For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
18002 As well as a truth value, propositions have a range of significance for their variables [Russell]
18003 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
21341 Philosophers of logic and maths insisted that a vocabulary of relations was essential [Russell, by Heil]
18246 Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
19157 Russell said the proposition must explain its own unity - or else objective truth is impossible [Russell, by Davidson]