 Second-order comprehension

Second-order comprehension is a scheme asserting the existence of a second-order relation variable whose members are exactly the tuples satisfying a permitted formula:Here $R$ is not free in $\phi$, and the displayed individual and relation parameters are universally closed in each instance. Under full semantics for second-order logic, the selected external subset of $D^k$ is always available. Under Henkin semantics, the selected subset must lie in ${\mathcal{H}}_k$ for every allowed assignment of parameters. Different systems permit different formula classes: for example, arithmetical comprehension theory restricts comprehension to formulas without bound set variables, while still permitting set parameters. Consequently, the exact formula and parameter policy must accompany any claim that a restricted range satisfies second-order comprehension.