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Ideas of Richard Dedekind, by Text
[German, 1831 - 1916, Born and died at Brunswick. Taught mathemtics in Zurich and Brunswick.]
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1872
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Continuity and Irrational Numbers
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Intro
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p.2
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17611
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We want the essence of continuity, by showing its origin in arithmetic
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§1
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p.4
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17612
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Arithmetic is just the consequence of counting, which is the successor operation
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§4
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p.15
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10572
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A cut between rational numbers creates and defines an irrational number
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p.27
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p.263
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18087
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If x changes by less and less, it must approach a limit
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1888 Jan
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p.173
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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
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1888
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Nature and Meaning of Numbers
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p.13
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10090
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Dedekind defined the integers, rationals and reals in terms of just the natural numbers [George/Velleman]
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p.21
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9153
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Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Fine,K]
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p.45
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9979
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Dedekind has a conception of abstraction which is not psychologistic [Tait]
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p.71
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14437
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Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Russell]
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p.88
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22289
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Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Potter]
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p.99
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18094
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Dedekind says each cut matches a real; logicists say the cuts are the reals [Bostock]
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p.101
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18096
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Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Bostock]
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p.116
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7524
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Order, not quantity, is central to defining numbers [Monk]
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p.124
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13508
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Dedekind gives a base number which isn't a successor, then adds successors and induction [Hart,WD]
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p.146
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9189
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Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dummett]
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p.200
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17452
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Ordinals can define cardinals, as the smallest ordinal that maps the set [Heck]
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p.248
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14130
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Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Russell]
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p.251
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14131
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Dedekind's ordinals are just members of any progression whatever [Russell]
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p.267
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18841
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Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt]
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Pref
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p.31
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9823
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Numbers are free creations of the human mind, to understand differences
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Pref
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p.32
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9824
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In counting we see the human ability to relate, correspond and represent
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§3 n13
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p.581
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8924
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Dedekind originated the structuralist conception of mathematics [MacBride]
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§64
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p.376
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10183
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An infinite set maps into its own proper subset [Reck/Price]
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2-3
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p.23
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10706
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Dedekind originally thought more in terms of mereology than of sets [Potter]
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I.1
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p.44
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9825
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A thing is completely determined by all that can be thought concerning it
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no. 66
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p.83
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22288
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We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Potter]
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V.64
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p.63
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9826
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A system S is said to be infinite when it is similar to a proper part of itself
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VI.73
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p.68
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9827
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We derive the natural numbers, by neglecting everything of a system except distinctness and order
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