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Single Idea 14128

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers ]

Full Idea

It is claimed that ordinals are prior to cardinals, because they form the progression which is relevant to mathematics, but they both form progressions and have the same ordinal properties. There is nothing to choose in logical priority between them.

Gist of Idea

Some claim priority for the ordinals over cardinals, but there is no logical priority between them

Source

Bertrand Russell (The Principles of Mathematics [1903], §230)

Book Ref

Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.241

A Reaction

We have an intuitive notion of the size of a set without number, but you can't actually start counting without number, so the ordering seems to be the key to the business, which (I would have thought) points to ordinals as prior.

Related Idea

Idea 14129 Ordinals presuppose two relations, where cardinals only presuppose one [Russell]

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]
14141 Ordinals are defined through mathematical induction [Russell]
14142 Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell]
14139 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [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]
14153 In geometry, empiricists aimed at premisses consistent with experience [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]
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]
14168 Occupying a place and change are prior to motion, so motion is just occupying places at continuous times [Russell]
14169 The 'universe' can mean what exists now, what always has or will exist [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]