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Let $\mathcal L_3 = \{<,c_o,c_1,\ldots\}$, where $c_o, c_1,\ldots$ are constant symbols. Let $T_3$ be the theory of DLO with sentences added asserting $c_o < c_1 < \ldots$.

I would like to show that $T_3$ has three countable models up to isomorphism. They can be constructed as follow:

  1. $\mathbb Q$, with $c_n = n$ for all $n \in \mathbb N$, so any sequence $c_0,c_1,\ldots$ are unbounded.
  2. $\mathbb Q$, with $c_n = -\frac{1}{n+1}$ for all $n \in \mathbb N$, so that the sequence has a limit in $\mathbb Q$.
  3. $\mathbb Q$ with $(c_n)$ equal to some increasing sequence of rational numbers converging to some irrational number, say $\sqrt{2}$. So that the sequence has a least upper bound.

Now let $\mathcal L_4 = \mathcal L_3 + \{P\}$ where $P$ is a unary predicate. Let $T_4$ be $T_3$ with added sentence

$$\forall x \forall y (x < y \rightarrow \exists z \exists w (x < z < y \wedge x , w < y \wedge P(z) \wedge \neg P(w)))$$

I would like to show $T_4$ is complete theory with four countable models.

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HINT. The 4 non isomorphic models are as follows:

a. As in your 1;

b. As in your 2 with $P(0)$;

c. As in your 2 with $\neg P(0)$;

d. As in your 3.

To prove completeness I would first adapt Cantor back-and-forth argument to prove that the theory without the constants is $\omega$-categorical and with elimination of quantifiers. Adding the constants preserve quantifier-elimination and completeness.

EDIT: I implicitely added to your $T_4$ the axioms $P(c_i)$ for all $i\in\omega$. (Or any other axioms that fix the "color" of the constants.) Otherwise there are $2^\omega$ complete theories extending your $T_4$. (Each with 4 non isomorphic models.)

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