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Full Idea
A Dedekind Cut is a division of rationals into two set (A1,A2) where every member of A1 is less than every member of A2. If n is the largest A1 or the smallest A2, the cut is produced by n. Some cuts aren't produced by rationals.
Gist of Idea
Cuts are made by the smallest upper or largest lower number, some of them not rational
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 5.4)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.171
| 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] |