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
Source
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]