Question from Marker's book T3 has three models up to isomorphism

Question

This question is from Marker's book.

Let $ \mathcal L_3 = \left\{ {< ,c_0,c_1, \dots}\right\} $ where $c_0,c_1, \dots$ are constants symbols. Let $T_3$ be the theory of dense linear orders without endpoints with sentences added asserting $c_0 < c_1 < \dots$

Show that $T_3$ has three countable models up to isomorphism.

[Hint: Consider the questions: Does $c_0,c_1,c_2 \dots$ have upper bound ? A least upper bound ?]

Here I can't see the connection between hint and question and I know the Real numbers and the Hyperreal numbers are models for $T_3$. I need some clue about the hint.

Answer 1

Hint: Use a back-and-forth argument to show that a countable model of $T_3$ is isomorphic to one of:

• $\mathbb{Q}$, with $c_n = n$ for all $n\in \mathbb{N}$,
• $\mathbb{Q}$, with $c_n = -\frac{1}{n+1}$ for all $n \in \mathbb{N}$,
• $\mathbb{Q}$, with $(c_n)$ equal to a particular increasing sequence of rational numbers converging to $\sqrt{2}$ (which particular sequence doesn't matter).

depending on whether $c_0, c_1, \ldots$ is unbounded, has a least upper bound, or has an upper bound but not a least upper bound, respectively.