Number of order preserving bijections on the set of integers

Question

Let $A$ be the set of all order preserving bijections on the set of integers

Question. What is the cardinality of $A$?

Thoughts. For every integer $r$ $$f(n)=n+r$$ is an order preserving bijection. Hence $A$ is at least countably infinite. But I can't prove that it is not uncountable.

Answer 1

Hint: Let $f \colon \mathbb Z \to \mathbb Z$ be an order preserving bijection. Let $r = f(0)$. Show that $f(x) = x + r$ for all $x \in \mathbb Z$.

(You can - for example - show this by two inductions on $\mathbb Z^+$ and on $\mathbb Z^-$.)