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 |
p.116 | 7524 | Order, not quantity, is central to defining numbers |
p.124 | 13508 | Dedekind gives a base number which isn't a successor, then adds successors and induction |
p.146 | 9189 | Dedekind said numbers were abstracted from systems of objects, leaving only their position |
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 |
§64 | p.376 | 10183 | An infinite set maps into its own proper subset |
2-3 | p.23 | 10706 | Dedekind originally thought more in terms of mereology than of sets |
I.1 | p.44 | 9825 | A thing is completely determined by all that can be thought concerning it |
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 |