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Single Idea 10572

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts ]

Full Idea

Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut.

Gist of Idea

A cut between rational numbers creates and defines an irrational number

Source

Richard Dedekind (Continuity and Irrational Numbers [1872], §4)

Book Ref

Dedekind,Richard: 'Essays on the Theory of Numbers' [Dover 1963], p.15

A Reaction

Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573.

Related Idea

Idea 10573 Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K]

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]
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]
10090 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
17452 Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
14131 Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
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]
18841 Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
14130 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
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]
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]