For each $n>1$ we shall construct a first-order theory $T_n$ with exactly n countable models.
Let $n>1$, consider the language $L_n=\left\{{R,c_1,…,c_n}\right\}$, where $R$ is a binary relation symbol, and each $c_n$ is a distinct constant symbol.
Put $$\phi=\forall{x}\forall{y}(x\neq{c_1}\wedge{y\neq{c_2,…c_n}}\longrightarrow{¬Rxy})$$ and
$$\psi=\forall{x}\forall{y}\forall{z}(x\neq{y}\wedge{x\neq{z}\wedge{y\neq{z}\wedge{Rxy}}}\longrightarrow{¬Rxz}).$$
Set $$T_n=\left\{{c_i\neq{c_j}:i\neq{j}}\right\}\cup\left\{{\phi,\psi}\right\}.$$
It is clear that $T_n$ is consistent, and thus by Lowenheim-Skolem theorem $T_n$ has a countable model.
Let $\mathcal{M}$ be a countable model of $T_n$, then since $\mathcal{M}\models{\phi}$ we must have that either $R^{\mathcal{M}}=\emptyset$ or $R^{\mathcal{M}}=\left\{{(c_1,c_{n_1}),…,(c_1,c_{n_m})}\right\}$, but since $\mathcal{M}\models{\psi}$ we get that if $R^{\mathcal{M}}\neq{\emptyset}$, then $\left |{R^{\mathcal{M}}}\right |=1$.
Therefore we must have that either $R^{\mathcal{M}}=\emptyset$ or $R^{\mathcal{M}}=\left\{{(c_1,c_m)}\right\}$ for some $m\leq{n}$.
It is clear that if $\mathcal{M}$ and $\mathcal{N}$ are models of $T_n$ with $R^{\mathcal{M}}=\emptyset$ and $R^{\mathcal{N}}=\emptyset$, then $\mathcal{M}\simeq{\mathcal{N}}$.
And also if $R^{\mathcal{M}}=\left\{{(c_1,c_i)}\right\}$ and $R^{\mathcal{N}}=\left\{{(c_1,c_j)}\right\}$, then $\mathcal{M}\simeq{\mathcal{N}}$ if and only if $i=j$, for
$\mathcal{M}\models{c_i\neq{c_j}}$ for $i,j\in{\left\{{2,…,n}\right\}}$ with $i\neq{j}$
By Lowenheim-Skolem theorem each case has a countable model.
Thus $T_n$ has exactly $n$ countable models.
I would like to see others examples.
Thanks
Best Answer
While this is not mandated in the original question, it is more interesting to ask for which finite $n$ there exists a complete theory with exactly $n$ countable models up to isomorphism.
For $n=1$, we can take the theory of dense linear orders (or any other $\omega$-categorical theory, for that matter).
For $n\ge3$, there is an elegant construction due to Ehrenfeucht. Let $D_k$ ($k\ge1$) be the theory of densely $k$-coloured linear orders: that is, its language consists of a binary predicate $<$, and unary predicates $P_1,\dots,P_k$, and its axioms state that $<$ is a linear order without end-points, each element satisfies exactly one of the predicates $P_1,\dots,P_k$, and for each $i=1,\dots,k$, the set of elements satisfying $P_i$ is dense. Using a similar zig-zag argument as for dense linear orders, it is easy to show that $D_k$ is complete, $\omega$-categorical, and it has elimination of quantifiers.
Now, let $T_{k+2}$ be an extension of $D_k$ in a language with extra constants $\{c_n:n\in\omega\}$, and axioms $\{c_n<c_m:n<m\}\cup\{P_1(c_n):n\in\omega\}$. By quantifier elimination for $D_k$, the theory $T_{k+2}$ is complete. Let $A,B$ be countable models of $T_{k+2}$. We can construct a partial isomorphism by mapping $c_n^A$ to $c_n^B$, each interval $(c_n,c_{n+1})_A$ onto $(c_n,c_{n+1})_B$, and similarly for the unbounded interval $(-\infty,c_0)$: indeed, all these intervals are countable models of $D_k$, hence they are isomorphic. Conversely, any isomorphism of $A$ and $B$ must look like that. Thus, the isomorphism type of $A$ is uniquely determined by the isomorphism type of the “remainder” $$A^*=\{a\in A:\forall n\in\omega\,c_n^A<a\}.$$ This remainder may be empty, or it may be nonempty with no least element (in which case it is the unique countable model of $D_k$), or it may have a least element $a$: in the last case, $(a,\infty)$ is the unique countable model of $D_k$, and the only choice we have is the colour of $a$. Thus, in total, $T_{k+2}$ has exactly $k+2$ countable models up to isomorphism.
Curiously, a theorem of Vaught states that the remaining case, $n=2$, is in fact impossible: there is no complete theory with exactly two countable models.
An outline of the proof is as follows. If $T$ has only two countable models, it has only countably many complete $n$-types for every $n<\omega$, hence it has an atomic model $A$, and a countable saturated model $S$. Since $T$ is not $\omega$-categorical, it has a nonprincipal complete $n$-type $p(\vec x)$ for some $n$. First, notice that $p$ is realized in $S$ but not in $A$, hence $A\not\simeq S$, hence every countable model of $T$ is isomorphic to $A$ or to $S$. Consider the theory $T'=T+p(\vec c)$, where $\vec c$ are new constants. If $M,N$ are countable models of $T'$, their reducts to the language of $T$ must be isomorphic to $S$, hence they are saturated (this property is preserved by expansion with extra constants). Since $T'$ is complete, this implies $M\simeq N$. Thus, $T'$ is $\omega$-categorical, and as such it has only finitely many complete $n$-types for every $n$. But then the same must hold for $T$, a contradiction.