Is there an ordered field whose natural/integer/rational part is definable?

Question

Does there exist an ordered field $(F;+,-,*,0,1,<)$, which is not $\mathbb{Q}$, such that its natural part and/or its integer part and/or its rational part is definable without parameters? By natural part, I mean the set $\{0,1,1+1,1+1+1,1+1+1+1,...\}$. By integer part, I mean the union of the natural part with the negatives of the natural part. By rational part, I mean the quotient of the integer part with itself.

Answer 1

Yes, as Atticus mentioned, in number fields you can always define the natural numbers. Check out theorem 3 page 208 of Rumely, Undecidability and Definability for the Theory of Global Fields . The idea is that you can quantify over finite sets in number fields, and then you can encode the natural numbers via saying that $n$ is a natural number iff there is a finite set that includes $0$ and whenever any $t$ is in the finite set, then either $t=n$ or $t+1$ is also in the finite set. Originally this is a theorem by Julia Robinson. Then just take any real number field and you are done.