In the situation I have an infinite set of closed wfs $\mathfrak{B}=\{\mathfrak{B}_{1}, \mathfrak{B}_{2}, \mathfrak{B}_{3}, ...\}$. It is also the case that no $\mathfrak{B}_{k}$ can be proven with the conjunction of all $\mathfrak{B}_{j}$ where $j < k$.
$T$ is the theory of all sentences that can be proven from $\mathfrak{B}$, and so prove that the theory $T$ is not finitely axiomatizable.
Naively, I understand this to be because there can be no model for every finite subset of $\mathfrak{B}$, so there is no model for $\mathfrak{B}$. In other words, not every theorem of $\mathfrak{B}$ holds as $\mathfrak{B}_{1},\mathfrak{B}_{2},\mathfrak{B}_{3}, ..., \mathfrak{B}_{j} \vdash \mathfrak{B}_{k}$ does not hold. Is this along the right lines?
This is an application of compactness. The single sentence $\mathfrak A$ axiomatizing $T$ must be a logical consequence of $\{\mathfrak B_1,\mathfrak B_2,\ldots\}$ and thus of finitely many $\mathfrak B_1,\ldots,\mathfrak B_n$. But it also must imply $\mathfrak B_{n+1},$ since that is a consequence of $\{\mathfrak B_1,\mathfrak B_2,\ldots\}.$ So then $\mathfrak B_{n+1}$ is a logical consequence of $\{\mathfrak B_1,\ldots,\mathfrak B_n\},$ contrary to assumption.