TITLE: Infinite linear (non-well) orderings
QUESTION [6 upvotes]: I can rather easily imagine the infinite linear (non-well) orderings $(\mathbb{Z},\leq)$, $(\mathbb{Q},\leq)$, $(\mathbb{R},\leq)$, each one of its own order type.
Are there "essentially" other infinite linear (non-well) orderings, next to or between the three above? 
If not so: Why not?
If so: (How) can they be imagined? Examples?

Where are they "located"? 
Are there any "between" $\mathbb{Z}$ and $\mathbb{Q}$?
Are there any "between" $\mathbb{Q}$ and $\mathbb{R}$? 
Or are they all "beyond" $\mathbb{R}$?
(What) does the latter have to do with the continuum hypothesis?
Or are there other "locations"?

REPLY [3 votes]: First, note that since there is a bijection between $\mathbb{Q}$ and $\mathbb{Z}$, you can endow the set $\mathbb{Q}$ with the order type of the usual $(\mathbb{Z}, <)$ and visa-versa. To do this, suppose $f : \mathbb{Q} \rightarrow \mathbb{Z}$ is a fixed bijection. You can define a new well-ordering on the set $\mathbb{Q}$ as follows: $q_1 \prec q_2$ if and only if $f(q_1) < f(q_1)$, where $<$ is the usual ordering on $\mathbb{Z}$. Hence $(\mathbb{Q}, \prec)$ is an ordering on $\mathbb{Q}$ that is isomorphic to $(\mathbb{Z}, <)$. 
From the above, you can see that it is not clear what "between" $\mathbb{Q}$ and $\mathbb{Z}$. Well-orderings can be compared since you can prove that any two well ordering $W_1$ and $W_2$, one can embed in the other. 
For your question of whether essentially other infinite linear order, the answer is yes. The usual way to say that a two linear ordering are the same is if there exists an order-preserving bijection between the two linear ordering. The example of $\mathbb{Q}$ above indicates that $\mathbb{Q}$ has at least two different order type: the usual one and an order type that is isomorphic to $(\mathbb{Z}, <)$. There are some very simple ways to construct even more non-isomorphic linear ordering. For example $\mathbb{Z}$ followed by another copy of $\mathbb{Z}$, with the ordering that all elements from the first copy is larger than all element from the second. Within the same copy of $\mathbb{Z}$, it would possess the usual ordering. 
The continuum hypothesis is about cardinals. $\aleph_1$ denotes the least infinite ordinal (transitive well-ordered set) that is not in bijection with any previous ordinal. The continuum hypothesis asserts whether $\aleph_1$ is in bijection with the real numbers.