In group theory, do $ \langle\mathbb{Z}, +\rangle $ and $ \langle\mathbb{R}, +\rangle $ have the same order?

Question

Of course, $ \mathbb{Z} $ is countable, while $ \mathbb{R}$ is uncountable, so the two cannot be isomorphic. However, the reason for my question is the notation for the order of $ $$ \langle\mathbb{Z}, +\rangle $$ $ that I encountered recently, which said $ order($$ \langle\mathbb{Z}, +\rangle) $$ = \infty$.

If that notation is correct, my assumption is that $ \langle\mathbb{Z}, +\rangle $ and $ \langle\mathbb{R}, +\rangle $ have the same order.

Answer 1

If $G$ is a group then the order $\operatorname{ord}(g)$ of an element $g\in G$ is the smallest positive integer $n$ with $g^n=1$, or one (symbolically) writes $\operatorname{ord}(g)=\infty$ if no such integer exists. The order of the group itself however is simply the cardinality of the underlying set. As such, one should never write $\infty$ for it, but rather $\aleph_0$ or $\mathfrak c$ or similar. Nevertheless, $\operatorname{ord}(G)<\infty$ to express that $G$ is a finite group is often found and might be acceptable.

Answer 2

Saying that the order of $\mathbb{Z}$ is $\infty$ is just saying that $\mathbb{Z}$ is not finite. As you have already noted, $\mathbb{R}$ and $\mathbb{Z}$ are not finite in different ways (countable vs. uncountable).