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

[numbers relating to position rather than total]

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