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Single Idea 10861

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

Full Idea

With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals.

Gist of Idea

Beyond infinity cardinals and ordinals can come apart

Source

Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13)

Book Ref

Clegg,Brian: 'Infinity' [Robinson 2003], p.169

The 30 ideas with the same theme [numbers relating to position rather than total]:

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]
9971 Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
9892 Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
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 the 'nth' instance [Russell]
14141 Ordinals are defined through mathematical induction [Russell]
14139 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [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]
22716 Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone]
17905 Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine]
13463 There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
13459 The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
13491 The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
13492 Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
10860 An ordinal number is defined by the set that comes before it [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]
17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
17760 Two infinite ordinals can represent a single infinite cardinal [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]