The notation ${\mathcal L}^\alpha_{\beta,\gamma}$ refers to the set of sentences of predicate calculus with less than $\alpha$ variables, conjunctions of size less than $\beta$, and quantification over families of less than $\gamma$ variables. Formally, the following are formulas of ${\mathcal L}^\alpha_{\beta,\gamma}$ in addition to the atomic formulas:

- $\neg\phi$ if $\phi\in {\mathcal L}^\alpha_{\beta,\gamma}$
- $\underset{i\in I}\bigvee \Phi_i$ if $|I|<\beta$ and $(\forall i\in I)\Phi_i\in {\mathcal L}^\alpha_{\beta,\gamma}$ and $NV(\underset{i\in I}\bigvee \Phi_i)<\alpha$
- $(\exists \{x_i\}_{i\in I})\phi$ if $|I|<\gamma$ and $\phi\in {\mathcal L}^\alpha_{\beta,\gamma}$

Where $NV(\phi)$ is the number of variables (free or bound) used anywhere in $\phi$.

Let $\lceil\gamma\rceil$ be the least limit ordinal greater than or equal to $\gamma$. Observe that ${\mathcal L}^\alpha_{\beta,\gamma}={\mathcal L}^\alpha_{\beta,\lceil\gamma\rceil}$.

So, my question is: why $\gamma$? More specifically, when is

$$ {\mathcal L}^\alpha_{\beta,\gamma}\neq {\mathcal L}^{min(\alpha,\lceil\gamma\rceil)}_{\beta,\infty} $$

I think they are the same, but if that is the case I can't see why the notation hasn't been reduced to simply ${\mathcal L}^\alpha_\beta$.

Note that there are various notions of what a "proof in ${\mathcal L}^\alpha_{\beta,\gamma}$" might mean, but I'm only interested in knowing if the notational distinction matters for contexts in which $\vdash$ means "true in all models".