TITLE: Is $ACA_0$ + `True Arithmetic exists' interpretable in $ACA$?
QUESTION [7 upvotes]: Maybe someone here can help me with a question concerning second-order arithmetic. Consider the system $ACA_T := ACA_0 + \exists X \forall x (x \in X \leftrightarrow T(x))$, where $T(x)$ is a $\Pi_1^1$ truth definition for first-order arithmetic (as given e.g. in Takeuti, Proof Theory, p. 183-188). So $ACA_T$ is $ACA_0$ plus a statement that says that the set of (Gödel numbers of) of true arithmetic exists. The question I have is: Is $ACA_T$ interpretable in $ACA$? I know that one can show that the arithmetical consequences of $ACA_T$ are precisely the consequences of $ACA$. But I would like to know whether $ACA_T$ can be directly interpreted in $ACA$.

REPLY [8 votes]: $\def\aca{\mathit{ACA}}\aca_T$ is finitely axiomatized, hence whenever it is interpretable in some theory, it is also interpretable in its finite subtheory. On the other hand, $\aca$ has full induction, hence it is (uniformly essentially) reflexive, that is, it proves the consistency of all its finite subtheories. It follows that if $\aca$ interpreted $\aca_T$, it would actually prove the consistency of $\aca_T$. (The converse also holds.)
Thus, assuming that your claim that $\aca$ and $\aca_T$ have the same arithmetical consequences is correct, the answer is negative, as $\aca_T$ does not prove its own consistency by Gödel’s theorem.