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

TL;DR

This paper constructs models of ZF set theory demonstrating the independence of various choice principles and comprehension axioms, highlighting the nuanced role of parameters and projective levels in these logical frameworks.

Contribution

It introduces generalized Jensen forcing iterations to build models that separate different choice and comprehension hypotheses, advancing understanding of their logical independence.

Findings

Models show certain choice principles are independent of ZF.

Parameter and projective level distinctions affect the implications of choice axioms.

Separation results for comprehension schemas in second-order arithmetic.

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 $\text{AC}_\omega(\varSigma^1_{\infty}).$