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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 Letter to Weber
 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]
 2-3 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