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

Abstract

We make use of generalized iterations of Jensen forcing to define a cardinal-preserving generic model of ZF for any $n\ge 1$ and each of the following four Choice hypotheses: (1) $\text{DC}(\mathbf\Pi^1_n)\land\neg\text{AC}_\omega(\varPi^1_{n+1})\,;$ (2) $\text{AC}_\omega(\text{OD})\land\text{DC}(\varPi^1_{n+1})\land \neg\text{AC}_\omega(\mathbf\Pi^1_{n+1});$ (3) $\text{AC}_\omega\land\text{DC}(\mathbf\Pi^1_n)\land\neg\text{DC}(\varPi^1_{n+1});$ (4) $\text{AC}_\omega\land\text{DC}(\varPi^1_{n+1})\land\neg\text{DC}(\mathbf\Pi^1_{n+1}).$ Thus if ZF is consistent and $n\ge1$ then each of these four conjunctions (1)--(4) is consistent with ZF. As for the second main result, let PA$^0_2$ be the 2nd-order Peano arithmetic without the Comprehension schema $\text{CA}$. For any $n\ge1$, we define a cardinal-preserving generic model of ZF, and a set $M\subseteq\mathcal P(\omega)$ in this model, such that $\langle\omega, M\rangle$ satisfies (5) PA$^0_2$ + $\text{AC}_\omega(\varSigma^1_{\infty})$ + $\text{CA}(\mathbf\Sigma^1_{n+1})$ + $\neg\text{CA}(\mathbf\Sigma^1_{n+1})$. Thus $\text{CA}(\mathbf\Sigma^1_{n+1})$ does not imply $\text{CA}(\mathbf\Sigma^1_{n+2})$ in PA$^0_2$ even in the presence of the full parameter-free (countable) Choice $\text{AC}_\omega(\varSigma^1_{\infty}).$