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Single Idea 13508
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
]
Full Idea
Dedekind's natural numbers: an object is in a set (0 is a number), a function sends the set one-one into itself (numbers have unique successors), the object isn't a value of the function (it isn't a successor), plus induction.
Gist of Idea
Dedekind gives a base number which isn't a successor, then adds successors and induction
Source
report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William D. Hart - The Evolution of Logic 5
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.124
A Reaction
Hart notes that since this refers to sets of individuals, it is a second-order account of numbers, what we now call 'Second-Order Peano Arithmetic'.
The
23 ideas
from 'Nature and Meaning of Numbers'
10090
|
Dedekind defined the integers, rationals and reals in terms of just the natural numbers
[Dedekind, by George/Velleman]
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7524
|
Order, not quantity, is central to defining numbers
[Dedekind, by Monk]
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17452
|
Ordinals can define cardinals, as the smallest ordinal that maps the set
[Dedekind, by Heck]
|
14131
|
Dedekind's ordinals are just members of any progression whatever
[Dedekind, by Russell]
|
14437
|
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil
[Dedekind, by Russell]
|
18094
|
Dedekind says each cut matches a real; logicists say the cuts are the reals
[Dedekind, by Bostock]
|
22289
|
Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic
[Dedekind, by Potter]
|
13508
|
Dedekind gives a base number which isn't a successor, then adds successors and induction
[Dedekind, by Hart,WD]
|
18096
|
Zero is a member, and all successors; numbers are the intersection of sets satisfying this
[Dedekind, by Bostock]
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18841
|
Categoricity implies that Dedekind has characterised the numbers, because it has one domain
[Rumfitt on Dedekind]
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14130
|
Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer
[Dedekind, by Russell]
|
9153
|
Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects
[Dedekind, by Fine,K]
|
9189
|
Dedekind said numbers were abstracted from systems of objects, leaving only their position
[Dedekind, by Dummett]
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9979
|
Dedekind has a conception of abstraction which is not psychologistic
[Dedekind, by Tait]
|
9823
|
Numbers are free creations of the human mind, to understand differences
[Dedekind]
|
9824
|
In counting we see the human ability to relate, correspond and represent
[Dedekind]
|
8924
|
Dedekind originated the structuralist conception of mathematics
[Dedekind, by MacBride]
|
10183
|
An infinite set maps into its own proper subset
[Dedekind, by Reck/Price]
|
10706
|
Dedekind originally thought more in terms of mereology than of sets
[Dedekind, by Potter]
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9825
|
A thing is completely determined by all that can be thought concerning it
[Dedekind]
|
22288
|
We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available
[Dedekind, by Potter]
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9826
|
A system S is said to be infinite when it is similar to a proper part of itself
[Dedekind]
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9827
|
We derive the natural numbers, by neglecting everything of a system except distinctness and order
[Dedekind]
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