Second order arithmetic and autonomous computability

An autonomous process can be described within the second order arithmetic or within the closely related Zermelo–Fraenkel theory (without the powerset axiom), which is convenient. The following key result is proved: If some autonomous oracle gives rise to a model for the second-order arithmetic then it automatically gives a model for Zermelo–Fraenkel theory (without the powerset axiom). The latter is naturally interpreted on hereditarily countable sets which are easily representable by countable trees satisfying the chain condition. In general, every autonomous process can be described in Zermelo–Fraenkel theory (without the powerset axiom); moreover, the description is absolute for every oracle model. Hence, there is no autonomous process yielding a model for the complete theory of the second order arithmetic.