= Solution
A <first-order theory> in a fixed <first-order signature> is a set of <first-order sentences>. Under the closed-theory convention used in the question, it contains all its <semantic entailments>: $T\models\theta$ implies $\theta\in T$. One can equivalently begin with a set of axioms and take its closure under consequence.
A <complete theory> is a consistent <first-order theory> that decides every <first-order sentence> $\theta$ in its <first-order signature>:
$$
\boxed{T\models\theta\quad\text{or}\quad T\models\neg\theta.}
$$
Consistency ensures that exactly one of these alternatives holds.
A <decidable theory> is one for which an <algorithm> always terminates and determines whether a given <first-order sentence> is a consequence:
$$
\boxed{\{\theta:T\models\theta\}\text{ is decidable}.}
$$
If the <first-order theory> is already closed under consequence, this is its membership problem. It is a condition on the consequences, not merely on whether the chosen set of axioms has a decidable membership test.
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