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that restrict the number of variables in formulas. Also, generalized counting quantifiers that say "there exists infinitely many" are not expressible using a finite number of formulas in first-order logic.
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with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in this context they are a notational shorthand. However, they are interesting in the context of logics such as
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502:{\displaystyle {\begin{aligned}\exists ^{\geq 0}xPx&\leftrightarrow \top \\\exists ^{\geq k+1}xPx&\leftrightarrow \exists x(Px\land \exists ^{\geq k}y(y\neq x\land Py))\end{aligned}}}
289:{\displaystyle {\begin{aligned}\exists ^{=0}xPx&\leftrightarrow \neg \exists xPx\\\exists ^{=k+1}xPx&\leftrightarrow \exists x(Px\land \exists ^{=k}y(y\neq x\land Py))\end{aligned}}}
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27:Mathematical logical term
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513:See also
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