The standard (a standard) definition of the reals is: let r be some subset of Q, the set of rationals. r is called a Dedekind cut if:

  • r is closed downwards; that is, if x ∈ r and yx, then y ∈ r.
  • r has no largest element; for every x ∈ r, there exists a y ∈ r such that y > x.

R, the set of real numbers, is then precisely { r ∈ P(Q) | r is a Dedekind cut }.

We embed Q in R by saying q_r (the embedding of q ∈ Q into R) = { x ∈ Q | x < q }. Then, a real is irrational iff it is not the embedding of a rational number int R. This is equivalent to saying: a Dedekind cut r is irrational iff its complement Q \ r has no smallest element.