$\mathbb{Q+Z}$ not isomorphic to $\mathbb{Q+N}$

Question

How can i prove that $\mathbb{Q+Z}$ not isomorphic to $\mathbb{Q+N}$. Suppose that there is isomorphism My thought was to pick, for example, two zeroes from $\mathbb{Q+Z}$ like $0$ and $\bar{0}$, then there are infinitely many numbers between them. Then for two elements from $\mathbb{Q+N}$, for example, $a$ and $b$ there are $\mathbb{Q}$ elements between them. But $\mathbb{Q+Z}$ not isomorphic to $\mathbb{Q}$. Am i right?

Answer 1

$\Bbb Q+\Bbb Z$ contains an upper set which is isomorphic to $\Bbb Z$, whereas $\Bbb Q+\Bbb N$ doesn't. In fact, let $V\subseteq \Bbb Q+\Bbb N$ be an upper set. If $V\cap\Bbb Q\ne \emptyset$, then there are some $a,b\in V$ such that $\{v\in V\,:\, a<v<b\}$ is infinite. If $V\subseteq \Bbb N$, then $V$ is well-ordered. Either way, $V$ is not isomorphic to $\Bbb Z$.