Of the three properties that a set of sentence may have: "consistent", "complete", "is a theory", which pairs imply the third?
It is Exercise 1.4.42 in Hinman's Fundamentals of Mathematical Logic. I guess if a set of sentence $\Delta$ is a complete propositional theory, then it is consistent. As it is similar to Proposition 1.4.29 in this book:
Every finite consistent and complete set of sentences is consistent (has a model).
I tried to find a model of such $\Delta$ . I chose truth assignment $V$ :
$$ V(p_n) = \begin{cases} \mathrm{T},p_n\in \Delta, \\\\ \mathrm{F},\text{otherwise}. \end{cases} $$
To prove it is truly a model, I tried to prove $\varphi\in \Delta\iff V(\varphi) = \mathrm{T}$ by induction.
For atomic sentences it is easy to prove. For $\psi = \neg \varphi$, as $\Delta$ is a complete set,
$$ \neg \psi \in \Delta \iff \psi \notin \Delta \iff V(\psi)\ne \mathrm{T} \iff V(\neg \psi)= \mathrm{T} $$
Then the problem goes to $\varphi\lor \psi$ . I don't know how to continue by using "$\Delta$ is a theory".