Does the Cantor Set with Irrational Endpoints Contain Rationals?

Question

1. Let $[a,b]$ be a nonempty interval with irrational endpoints.
2. Choose distinct irrational points $p,q\in(a,b)$
3. Remove the subinterval $(p,q)$ from the initial interval $[a,b]$
4. Repeat the process similar to the Cantor set process, choosing irrational endpoints at each stage.

Does the resultant set contain rational points?