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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]
| 10572 | A cut between rational numbers creates and defines an irrational number [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] |
| 14436 | A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell] |
| 18248 | A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro] |
| 15274 | Points are 'continuous' if any 'cut' point participates in both halves of the cut [Harré/Madden] |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| 10575 | Why should a Dedekind cut correspond to a number? [Fine,K] |
| 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |