Can the number of countable models of a complete first-order theory decrease after adding constants?

I believe that the number of models can indeed decrease. The following seems to be an example of $5 \to 3$:

Start with $T$ the model companion of the theory of "valued trees": the language has two sorts $M$ and $\Gamma$, $\Gamma$ is equipped with a linear order $\leq_{\Gamma}$ and is a model of DLO. The sort $M$ has a tree structure, say in the language $\{\leq ,\wedge \}$, so $\leq $ is a partial order such that the set of predecessors of any point is a chain and $\wedge$ maps two points to their infimum. There is also a valuation map $v:M \to \Gamma$ which is increasing and such that for any $a\in M$, the restriction of $v$ to $\{x\in M:x<a\}$ is an injection onto $\{\gamma\in \Gamma:\gamma<v(a)\}$. This theory is $\aleph_0$-categorical.

Add a countable set of constants $\{c_0,c_1,\ldots,\}$ interpreted such that $c_0<c_1<\cdots$. Call $T_1$ the resulting theory. Then $T_1$ has 5 countable models:

• the prime model where the $c_i$'s are cofinal;
• two models with no element above all the $c_i$'s (one with a minimal valuation larger that all $v(c_i)$ and one without);
• two models with an element above all the $c_i$'s (same as the previous case).

If we now add another constant which is above all the $c_i$'s, then we have only 3 models (one where the constant is minimal above the $c_i$'s, one where it isn't, but there is such a minimal point, one where there is no such minimal point).

Edit: Example of $2^{\omega}\to_{\omega} 3$:

We modify a little bit the example above. Take the constants $c_i$ to name a subtree that has exactly $\aleph_0$ branches, all going all the way up in the tree. This has $2^{\aleph_0}$ many countable models, since we can independently add or not elements at the top of each branch. Now add $\aleph_0$ constants, one above each branch of the named tree and impose that they have the same valuation. If I am not mistaken, this has now only 3 countable models similarly as above.

See the paper Nonnessential extensions of complete theories(B. Omarov)

Translated from Algebra i Logika, Vol. 22, No. 5, pp. 542-550, September-October, 1983] Original article submitted June ii, 1982.

By adding constants (that may realize new types) Omarov write `A. D. Taimanov posed the following questions: "Is it possible to lower the number of countable models from continuum to countable , and from continuum to the finite number k ?" These questions are also answered affirmatively.