From "A Mathematical Introduction to Logic" (Enderton) excise 4.1.3
Let $ϕ$ be a formula in which only the n-place predicate variable $X$ occurs free. Say that an n-ary relation R on |A| is implicitly defined in $$ by $ϕ$ iff $ $ satisfies $ϕ$ with an assignment of $R$ $\to$ $X$ but does not satisfy $ϕ$ with an assignment of any other relation to $X$.
Show that # Th, the set of Gödel numbers of first-order sentences true in $$, is implicitly definable in by a formula without quantified predicate or function variables. Suggestion: The idea is to write down conditions that the set of true sentences must meet.
I did this by $ϕ$ says about $X$ that $X$ is a complete consistent theory that contains $A_E$(robinson arithmetic). However there are infinitely many such theories. How can I distinguish between such theories and number theory? Or is there any Method to Depict number theory?