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6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts

[defining real numbers by cutting the line of rationals]

11 ideas
A cut between rational numbers creates and defines an irrational number [Dedekind]
I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Russell]
A real number is the class of rationals less than the number [Korsgaard on Russell/Whitehead]
Points are 'continuous' if any 'cut' point participates in both halves of the cut [Harré/Madden]
For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
Dedekind says each cut matches a real; logicists say the cuts are the reals [Bostock]
Why should a Dedekind cut correspond to a number? [Fine,K]
Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]