I am trying to prove that there is no integer between $0$ and $1$. The standard proof uses well ordering principle. But if one takes integers as subset of rationals and rational numbers as an ordered field, by contradiction if there exists any integer $N$ (which is repeated addition of multiplicative identity $N$ times) between $0$ and $1$, then ordered field is failed.
Now the question is, does this mean that well ordering principle is someway followed when one defines an order on the field satisfying field structure? So a total order on a field satisfying field structure contains well ordering theorem? I am not bothered about real numbers as I am not using any completeness property