Full Idea
As Eudoxus claimed, two distinct real numbers cannot both make the same cut in the rationals, for any two real numbers must be separated by a rational number. He did not say, though, that for every such cut there is a real number that makes it.
Clarification
Eudoxus worked at Plato's Academy
Gist of Idea
For Eudoxus cuts in rationals are unique, but not every cut makes a real number
Source
David Bostock (Philosophy of Mathematics [2009], 4.4)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.98
A Reaction
This is in Bostock's discussion of Dedekind's cuts. It seems that every cut is guaranteed to produce a real. Fine challenges the later assumption.
Related Idea
Idea 10575 Why should a Dedekind cut correspond to a number? [Fine,K]