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Single Idea 13459
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
]
Full Idea
We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals.
Gist of Idea
The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers
Source
William D. Hart (The Evolution of Logic [2010], 1)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.26
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]
|
14139
|
Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic
[Russell]
|
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]
|
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]
|
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]
|
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]
|
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]
|
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]
|