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June 15, 2022
Rob Papen Predator V1.6.2c AU VST RTAS OSX Xdb.rar
Rob Papen Predator V1.6.2c AU VST RTAS OSX Xdb.rar
Q:
Are there FOL formulas whose free variables are restricted to proper subsets of free variables of the formula itself?
Let $T$ be a first-order theory. For example, $T$ could be Peano arithmetic (PA) or the theory of a singleton set (TC). Is there any way to characterize the size of the sets $\mathcal{F}(\phi)$ and $\mathcal{F}(\overline{\phi})$ for formulas $\phi$ and $\overline{\phi}$ in $T$, e.g., for bounded quantifiers?
I believe, though I don’t have a proof, that there is no such characterization possible in P.
A:
Let $T$ be any theory, and for each $\phi$ and $\overline\phi$ we define the subset of free variables of $\phi$ as
$$\mathcal{F}(\phi)=\{x\in\mathcal{F}(\overline{\phi})\,|\,\text{the same variable occurs in $\phi$ and $\overline{\phi}$} \}.$$
We could, of course, also define a similar subset $\mathcal{F}(\overline{\phi})$. To clarify what we mean, here are some non-empty subsets of free variables in Peano arithmetic:
$$\mathcal{F}(\text{$\exists x\,(x\!\!\!\!\!\!\pmod{1})$})=\{0,\pm1\},$$
$$\mathcal{F}(\text{$\exists x\,\forall y\,(x\!\!\!\!\!\!\pmod{1}=y)$})=\{0,\pm1,\pm2\},$$
$$\mathcal{F}(\text{$\forall y\,\exists x\,(x\!\!\!\!\!\!\pmod{1}=y)$})=\{0,\pm1\},$$
$$\mathcal{F}(\text{$\forall x\,\forall y\,(x\!\!\!\!\!\!\pmod{1}=
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