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 (and assuming I was not mistaken) 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 $|FV(\underset{i\in I}\bigvee \Phi_i)|<\alpha$
• $(\exists \{x_i\}_{i\in I})\phi$ if $|I|<\gamma$ and $\phi\in {\mathcal L}^\alpha_{\beta,\gamma}$

So, my question is: why $\gamma$? More specifically, if $\lceil\gamma\rceil$ is the least limit ordinal greater than or equal to $\gamma$, then 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 ${\mathcal L}^\alpha_\beta$. My reasoning goes like this:

• The $\gamma=0$ case is obvious; you get no quantifiers.
• All values $0<\gamma\leq\omega$ yield the same set of formulas because the set of formulas is closed under arbitrary composition (so if $\gamma=2$ you can simulate $\gamma=4$ by using nestings twice as deep)
• 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 $\omega < \alpha < \omega\cdot{2}$ 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?)

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".

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".

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

$$ L^\alpha_{\beta,\gamma}\neq \begin{cases} L^0_{\beta,\infty} & \text{\ if\ } 0=\gamma \\ L^{min(\alpha,\omega)}_{\beta,\infty} & \text{\ if\ } 0<\gamma\leq\omega \\ L^{min(\alpha,\gamma)}_{\beta,\infty} & \text{\ if\ } \omega<\gamma \\ \end{cases} $$

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:

• The $\gamma=0$ case is obvious; you get no quantifiers.
• All values $0<\gamma\leq\omega$ yield the same set of formulas because the set of formulas is closed under arbitrary composition (so if $\gamma=2$ you can simulate $\gamma=4$ by using nestings twice as deep)
• 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?)

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 (and assuming I was not mistaken) 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 $|FV(\underset{i\in I}\bigvee \Phi_i)|<\alpha$
• $(\exists \{x_i\}_{i\in I})\phi$ if $|I|<\gamma$ and $\phi\in {\mathcal L}^\alpha_{\beta,\gamma}$

So, my question is: why $\gamma$? More specifically, if $\lceil\gamma\rceil$ is the least limit ordinal greater than or equal to $\gamma$, then 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 ${\mathcal L}^\alpha_\beta$. My reasoning goes like this:

• The $\gamma=0$ case is obvious; you get no quantifiers.
• All values $0<\gamma\leq\omega$ yield the same set of formulas because the set of formulas is closed under arbitrary composition (so if $\gamma=2$ you can simulate $\gamma=4$ by using nestings twice as deep)
• 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 $\omega < \alpha < \omega\cdot{2}$ 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?)

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".