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Single Idea 17452
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
]
Full Idea
Dedekind and Cantor said the cardinals may be defined in terms of the ordinals: The cardinal number of a set S is the least ordinal onto whose predecessors the members of S can be mapped one-one.
Gist of Idea
Ordinals can define cardinals, as the smallest ordinal that maps the set
Source
report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 5
Book Ref
-: 'Notre Dame Journal of Formal Logic' [-], p.200
The
19 ideas
with the same theme
[which type of numbers is the most fundamental?]:
11044
|
One is prior to two, because its existence is implied by two
[Aristotle]
|
10091
|
God made the integers, all the rest is the work of man
[Kronecker]
|
10090
|
Dedekind defined the integers, rationals and reals in terms of just the natural numbers
[Dedekind, by George/Velleman]
|
17452
|
Ordinals can define cardinals, as the smallest ordinal that maps the set
[Dedekind, by Heck]
|
7524
|
Order, not quantity, is central to defining numbers
[Dedekind, by Monk]
|
9983
|
Cantor took the ordinal numbers to be primary
[Cantor, by Tait]
|
18256
|
Quantity is inconceivable without the idea of addition
[Frege]
|
13510
|
Could a number just be something which occurs in a progression?
[Russell, by Hart,WD]
|
14128
|
Some claim priority for the ordinals over cardinals, but there is no logical priority between them
[Russell]
|
14129
|
Ordinals presuppose two relations, where cardinals only presuppose one
[Russell]
|
14132
|
Properties of numbers don't rely on progressions, so cardinals may be more basic
[Russell]
|
13489
|
Von Neumann treated cardinals as a special sort of ordinal
[Neumann, by Hart,WD]
|
18255
|
Addition of quantities is prior to ordering, as shown in cyclic domains like angles
[Dummett]
|
9191
|
Ordinals seem more basic than cardinals, since we count objects in sequence
[Dummett]
|
13411
|
If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation
[Benacerraf]
|
9151
|
Benacerraf says numbers are defined by their natural ordering
[Benacerraf, by Fine,K]
|
18102
|
A cardinal is the earliest ordinal that has that number of predecessors
[Bostock]
|
13892
|
One could grasp numbers, and name sizes with them, without grasping ordering
[Wright,C]
|
8661
|
The natural numbers are primitive, and the ordinals are up one level of abstraction
[Friend]
|