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Single Idea 13508

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers ]

Full Idea

Dedekind's natural numbers: an object is in a set (0 is a number), a function sends the set one-one into itself (numbers have unique successors), the object isn't a value of the function (it isn't a successor), plus induction.

Gist of Idea

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

Source

report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William D. Hart - The Evolution of Logic 5

Book Ref

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.124

A Reaction

Hart notes that since this refers to sets of individuals, it is a second-order account of numbers, what we now call 'Second-Order Peano Arithmetic'.

The 28 ideas from Richard Dedekind

17611 We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
17612 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 If x changes by less and less, it must approach a limit [Dedekind]
18244 I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
9153 Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]
9979 Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait]
9189 Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett]
18841 Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
13508 Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
18096 Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
14130 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
14437 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
18094 Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
22289 Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter]
14131 Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
10090 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
17452 Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
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 A thing is completely determined by all that can be thought concerning it [Dedekind]
22288 We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter]
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]