Full Idea
Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio.
Clarification
'Commensurable' implies 'knowable'
Gist of Idea
The two sides of the Cut are, roughly, the bounding commensurable ratios
Source
Shaughan Lavine (Understanding the Infinite [1994], II.6)
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.38
A Reaction
Thus there is the problem of whether the contents of the gap are one unique thing, or many.