  Using a nonstandard model of Peano arithmetic, we show that there are
quasi-Euclidean subrings of Q[x] which are not k-stage Euclidean for any norm
and positive integer k. These subrings can be either PID or non-UFD, depending
on the choice of parameters in our construction. In both cases, there are
2^{\omega} such domains up to ring isomorphism.
