more from this thinker | more from this text
Full Idea
Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio.
Clarification
'Commensurable' implies 'knowable'
Gist of Idea
The two sides of the Cut are, roughly, the bounding commensurable ratios
Source
Shaughan Lavine (Understanding the Infinite [1994], II.6)
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.38
A Reaction
Thus there is the problem of whether the contents of the gap are one unique thing, or many.
| 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] |