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Single Idea 17905
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
]
Full Idea
The condition on an explication of number can be put succinctly: any progression will do nicely. Russell once held that one must also be able to measure multiplicity, but this was a mistake; any progression can be fitted to that further condition.
Gist of Idea
Any progression will do nicely for numbers; they can all then be used to measure multiplicity
Source
Willard Quine (Word and Object [1960], §54)
Book Ref
Quine,Willard: 'Word and Object' [MIT 1969], p.262
A Reaction
[compressed] This is the strongest possible statement that the numbers are the ordinals, and the Peano Axioms will define them. The Fregean view that cardinality comes first is redundant.
Related Idea
Idea 17906
To explain numbers you must also explain cardinality, the counting of things [Benacerraf]
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]
|
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]
|
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]
|
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]
|
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]
|