I'm a starter in learning model theory and found the following problem. But I'm not sure, how to solve it.
We have a language that consists of a single unary relation symbol $R$. Consider the theory $$T = \{ \tau_n := \exists x_1 \dots x_n (\bigwedge_{1 \leq i \leq n} R(x_i) \land \bigwedge_{1 \leq i < j \leq n} x_i \neq x_j) : n \in \mathbb{N} \}.$$ Find all possible semantic completions $T^\prime$ of $T$.
(An L-theory T is said to be semantically complete if for every L-sentence φ, T $\models$ φ or T $\models$ ¬φ. Call an L-theory $T^\prime \supseteq T$ a semantic completion if it is satisfiable and semantically complete.)
I'm afraid I did not get very far in solving this problem. $T$ itself seemed to be semantically complete, yet every sentence I tried to add to $T$ to obtain a semantic completion had a form which seemed to be equivalent to $\tau_n$.
Am I missing some kind of valuable tool here, or is this problem so easy that it hurts? Any tips are welcome!
The formula $\tau_n$ says that $R$ has at least $n$ elements.
The theory $T$ says that the set $R$ is infinite.
Incidentally, note that $T$ does not say anything about the complement of $R$.
For every $i\in\mathbb N \cup\{\infty\}$ there is a complete theory $T_i\supseteq T$ saying that $\neg R$ has exactly $i$ elements.
Can you axiomatize the theories $T_i$?
Can you prove that these are complete?
To prove that these are all the possible conpletion of $T$ pick any (countable) model of $T$ and prove that it models $T_i$ for some $i$.