Can someone show me how to adapt following formula to second order ? I mean that in SO formulas should begin with quantification over second order elements:
$\forall_{v\in V}\exists_{X\subseteq V} \phi$ where $\phi$ is first order formula, but $X$ is set, and $v$ is first order element.
Adapt formula in mix order (first order and second order) to second order
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1 Answers
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First of all, that's a perfectly valid second-order formula - there's no reason first-order quantifiers can't occur outside second-order quantifiers.
That said, in full second-order logic we can indeed get this into the form you want. Namely, by looking at binary relations! Basically, we want to look at a relation "$x$ is in the set corresponding to $v$".
Specifically, consider the formula $$\exists F\forall v\varphi^{ran},$$ where $\varphi^{ran}$ is the formula gotten from $\varphi$ by:
- replacing all subformulas of the form "$t\in X$" with "$F(v, t)$"
and
- replacing all subformulas of the form "$X=Y$" with "$\forall z(F(v, z)\iff z\in Y)$".
Intuitively, $\{x: F(v, x)\}$ is the $X$ corresponding to a given $v$ in the expression "$\forall v\exists X . . .$.
This has the desired properties.