Tarski's theorem, interpreted in Peano Arithmetic, says there is no predicate $T$ such that $PA\vdash T(\phi)\leftrightarrow \phi$. However, we know that there are partial truth predicates for each $k< \omega$ such that, for all $\phi \in \Sigma_k$, $PA\vdash T_k(\phi)\leftrightarrow \phi$. What is wrong with this supposed truth predicate, I will call $T_\omega$? I will define it by means of a recursive algorithm.
On input $\phi$, determine $k$ as the least $j$ such that $\phi\in\Sigma_j$. Then output $T_\omega(\phi) = T_k(\phi)$.
That's not actually an algorithm in the sense of a computable process: already, checking truth of $\Sigma_1$ sentences is not computable.
And if we switch to the language of, well, language, things aren't any better. Presumably the formula you have in mind is something like
Determining $k$ is of course easy. The problem is that you've essentially quantified over the $T_k$s. This can only be done if you can whip up a formula $T(x,y)$ where for each $k$ the formula $T(\underline{k}, y)$ corresponds to $T_k(y)$ ... but that's exactly what you're trying to do here.
Put another way, even if the sequence of formulas $(\psi_i)_{i\in\mathbb{N}}$ is as simple as you want (e.g. computable), expressions like $$\forall x(P(x)\rightarrow \psi_x(a))$$ are not first-order formulas: we can't have "variable formulas."