more from this thinker | more from this text
Full Idea
There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'.
Gist of Idea
A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII)
Book Ref
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.69
Related Idea
Idea 14437 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
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] |