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Single Idea 9826
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
]
Full Idea
A system S is said to be infinite when it is similar to a proper part of itself.
Clarification
We would now call a 'system' a 'set'
Gist of Idea
A system S is said to be infinite when it is similar to a proper part of itself
Source
Richard Dedekind (Nature and Meaning of Numbers [1888], V.64)
Book Ref
Dedekind,Richard: 'Essays on the Theory of Numbers' [Dover 1963], p.63
The
28 ideas
from Richard Dedekind
17611
|
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]
|
10572
|
A cut between rational numbers creates and defines an irrational number
[Dedekind]
|
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]
|
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]
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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
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Dedekind defined the integers, rationals and reals in terms of just the natural numbers
[Dedekind, by George/Velleman]
|
7524
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Order, not quantity, is central to defining numbers
[Dedekind, by Monk]
|
17452
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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]
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22289
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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]
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9189
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Dedekind said numbers were abstracted from systems of objects, leaving only their position
[Dedekind, by Dummett]
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9979
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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]
|
9825
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A thing is completely determined by all that can be thought concerning it
[Dedekind]
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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]
|
9827
|
We derive the natural numbers, by neglecting everything of a system except distinctness and order
[Dedekind]
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