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Single Idea 13530
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
]
Full Idea
Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.
Gist of Idea
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)
Book Ref
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.77
A Reaction
He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).
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]
|
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]
|
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]
|
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]
|
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]
|
17758
|
Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
[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]
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