The notation $L^\alpha_{\beta,\gamma}$ 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 (and assuming I was not mistaken) the following are formulas of $L^\alpha_{\beta,\gamma}$ in addition to the atomic formulas:

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

So, my question is: why $\gamma$? More specifically, when is $L^\alpha_{\beta,\gamma}\neq L^{min(\alpha,\gamma)}_{\beta,\infty}$?

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 $L^\alpha_\beta$. My reasoning goes like this:

• If $\alpha < \gamma$ no quantifier can actually quantify over $\alpha$ or more variables (unless some don't appear, in which case you can just delete them), since any formula which did so would involve a subformula with $\alpha$ or more free variables, which would be ill-formed..
• If $0 < \gamma < \alpha \leq \omega$ you can just nest finitely many $<\gamma$-variable quantifiers to achieve $<\alpha$-variable quantification, so increasing $\gamma$ to be equal to $\alpha$ will admit additional well-formed formulas, but each of them will be equivalent to some formula which had already been admitted.
• If $\gamma < \alpha$ and $\omega < \alpha$ then a formula with $\gamma$ or more free variables can never appear as a subformula of a closed formula (sentence) because no cardinal $<\gamma$ multiplied by a finite cardinal (nesting depth) is ever equal to a greater cardinal. So reducing $\alpha$ to equal $\gamma$ would not change the set of well-formed sentences (although I guess it might affect the set of formulas, but the sentences are what really matter, right?)