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Single Idea 17611
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
]
Full Idea
It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity.
Gist of Idea
We want the essence of continuity, by showing its origin in arithmetic
Source
Richard Dedekind (Continuity and Irrational Numbers [1872], Intro)
Book Ref
Dedekind,Richard: 'Essays on the Theory of Numbers' [Dover 1963], p.2
A Reaction
[He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut'].
The
28 ideas
from Richard Dedekind
17611
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We want the essence of continuity, by showing its origin in arithmetic
[Dedekind]
|
17612
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Arithmetic is just the consequence of counting, which is the successor operation
[Dedekind]
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10572
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A cut between rational numbers creates and defines an irrational number
[Dedekind]
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18087
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If x changes by less and less, it must approach a limit
[Dedekind]
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18244
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I say the irrational is not the cut itself, but a new creation which corresponds to the cut
[Dedekind]
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14437
|
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil
[Dedekind, by Russell]
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18094
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Dedekind says each cut matches a real; logicists say the cuts are the reals
[Dedekind, by Bostock]
|
13508
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Dedekind gives a base number which isn't a successor, then adds successors and induction
[Dedekind, by Hart,WD]
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18096
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Zero is a member, and all successors; numbers are the intersection of sets satisfying this
[Dedekind, by Bostock]
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18841
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Categoricity implies that Dedekind has characterised the numbers, because it has one domain
[Rumfitt on Dedekind]
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14130
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Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer
[Dedekind, by Russell]
|
10090
|
Dedekind defined the integers, rationals and reals in terms of just the natural numbers
[Dedekind, by George/Velleman]
|
7524
|
Order, not quantity, is central to defining numbers
[Dedekind, by Monk]
|
17452
|
Ordinals can define cardinals, as the smallest ordinal that maps the set
[Dedekind, by Heck]
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14131
|
Dedekind's ordinals are just members of any progression whatever
[Dedekind, by Russell]
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22289
|
Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic
[Dedekind, by Potter]
|
9153
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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]
|
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]
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10706
|
Dedekind originally thought more in terms of mereology than of sets
[Dedekind, by Potter]
|
9825
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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]
|
9826
|
A system S is said to be infinite when it is similar to a proper part of itself
[Dedekind]
|
9827
|
We derive the natural numbers, by neglecting everything of a system except distinctness and order
[Dedekind]
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