more from Richard Dedekind

Single Idea 10572

[catalogued under 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts]

Full Idea

Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut.

Gist of Idea

A cut between rational numbers creates and defines an irrational number


Richard Dedekind (Continuity and Irrational Numbers [1872], 4)

Book Reference

Dedekind,Richard: 'Essays on the Theory of Numbers' [Dover 1963], p.15

A Reaction

Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573.

Related Idea

Idea 10573 Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K]