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

A section on the Reflection principle is added in the last chapter. Using the results on DC we prove that Reflection is not finitely axiomatizable