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Ideas of Richard Dedekind, by Text
[German, 1831  1916, Born and died at Brunswick. Taught mathemtics in Zurich and Brunswick.]
1872

Continuity and Irrational Numbers

Intro

p.2

17611

We want the essence of continuity, by showing its origin in arithmetic

§1

p.4

17612

Arithmetic is just the consequence of counting, which is the successor operation

§4

p.15

10572

A cut between rational numbers creates and defines an irrational number

p.27

p.263

18087

If x changes by less and less, it must approach a limit

1888 Jan

p.173

18244

I say the irrational is not the cut itself, but a new creation which corresponds to the cut

1888

Nature and Meaning of Numbers


p.13

10090

Dedekind defined the integers, rationals and reals in terms of just the natural numbers [George/Velleman]


p.21

9153

Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Fine,K]


p.45

9979

Dedekind has a conception of abstraction which is not psychologistic [Tait]


p.71

14437

Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Russell]


p.88

22289

Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Potter]


p.99

18094

Dedekind says each cut matches a real; logicists say the cuts are the reals [Bostock]


p.101

18096

Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Bostock]


p.116

7524

Order, not quantity, is central to defining numbers [Monk]


p.124

13508

Dedekind gives a base number which isn't a successor, then adds successors and induction [Hart,WD]


p.146

9189

Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dummett]


p.200

17452

Ordinals can define cardinals, as the smallest ordinal that maps the set [Heck]


p.248

14130

Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Russell]


p.251

14131

Dedekind's ordinals are just members of any progression whatever [Russell]


p.267

18841

Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt]

Pref

p.31

9823

Numbers are free creations of the human mind, to understand differences

Pref

p.32

9824

In counting we see the human ability to relate, correspond and represent

§3 n13

p.581

8924

Dedekind originated the structuralist conception of mathematics [MacBride]

§64

p.376

10183

An infinite set maps into its own proper subset [Reck/Price]

23

p.23

10706

Dedekind originally thought more in terms of mereology than of sets [Potter]

I.1

p.44

9825

A thing is completely determined by all that can be thought concerning it

no. 66

p.83

22288

We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Potter]

V.64

p.63

9826

A system S is said to be infinite when it is similar to a proper part of itself

VI.73

p.68

9827

We derive the natural numbers, by neglecting everything of a system except distinctness and order
