more from this thinker | more from this text
Full Idea
Divide points into left and right set. They're 'continuous' if that point is either last member of left set, and greatest lower bound of right (so no least member), or least upper bound of left set (so no last member) and first member of right set.
Gist of Idea
Points are 'continuous' if any 'cut' point participates in both halves of the cut
Source
Harré,R./Madden,E.H. (Causal Powers [1975], 6.IV)
Book Ref
Harré,R/Madden,E.H.: 'Causal Powers: A Theory of Natural Necessity' [Blackwell 1975], p.111
A Reaction
The best attempt I have yet encountered to explain a Dedekind Cut for the layperson. I gather modern mathematicians no longer rely on this way of defining the real numbers.
Related Idea
Idea 10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
| 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] |