Let $\mathcal{L}= \{c,R,f \}$ a language where $c$ is a constant symbol, $R$ is a binary relation symbol and $f$ is a binary function symbol.
Let denote $\Bbb{R}^+:=\{r\in \Bbb{R} : r > 0\}$ and $\Bbb{Q}^+:=\{q\in \Bbb{Q} : q > 0\}$
Prove the following:
- $(\Bbb{R},0,<,+) \cong (\Bbb{R}^+,1,<,\cdot)$
- $(\Bbb{Q},0,<,+) \ncong (\Bbb{Q}^+,1,<,\cdot)$
I think that for the first the exponential $h(x)=e^x$ is a isomorphism because it is biyective between both sets and it is increasing, so it preserves the order. Despite, I am getting stuck with the first, I guess it exists a formula which $\Bbb{Q}$ satisfies and $\Bbb{Q}^+$ not or viceversa, but I don't find it. Possible ideas would be appreciated.