On the significance of parameters and the projective level in the Choice and Comprehension axioms

We make use of generalized iterations of Jensen forcing to define a cardinal-preserving generic model of ZF for any n≥ 1 and each of the following four Choice hypotheses: (1) DC(Π¹ₙ)_ω(¹ₙ₊₁)\,; (2) AC_ω(OD)(¹ₙ₊₁) _ω(Π¹ₙ₊₁); (3) AC_ω(Π¹ₙ)(¹ₙ₊₁); (4) AC_ω(¹ₙ₊₁)(Π¹ₙ₊₁). Thus if ZF is consistent and n≥1 then each of these four conjunctions (1)--(4) is consistent with ZF. As for the second main result, let PA⁰₂ be the 2nd-order Peano arithmetic without the Comprehension schema CA. For any n≥1, we define a cardinal-preserving generic model of ZF, and a set M⊆ P(ω) in this model, such that ω, M satisfies (5) PA⁰₂ + AC_ω(¹∞) + CA(Σ¹ₙ₊₁) + (Σ¹ₙ₊₁). Thus CA(Σ¹ₙ₊₁) does not imply CA(Σ¹ₙ₊₂) in PA⁰₂ even in the presence of the full parameter-free (countable) Choice AC_ω(¹∞).