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### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

#### [numbers relating to position rather than total]

29 ideas
 17258 If we just say one, one, one, one, we don't know where we have got to [Hobbes]
 14131 Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
 15911 Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
 9892 Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
 9971 Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
 8640 We cannot define numbers from the idea of a series, because numbers must precede that [Frege]
 14142 Ordinals are types of series of terms in a row, rather than than the 'nth' instance [Russell]
 14139 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell]
 14141 Ordinals are defined through mathematical induction [Russell]
 14145 For Cantor ordinals are types of order, not numbers [Russell]
 13487 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
 12336 A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou]
 17905 Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine]
 13459 The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
 13463 There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
 13492 Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
 13491 The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
 13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
 10860 An ordinal number is defined by the set that comes before it [Clegg]
 10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
 13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
 10680 The theory of the transfinite needs the ordinal numbers [Hossack]
 17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
 17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
 17760 Two infinite ordinals can represent a single infinite cardinal [Walicki]
 17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
 17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
 17928 Ordinal numbers represent order relations [Colyvan]
 18842 Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set [Rumfitt]