Ideals and strong axioms of determinacy

Abstract:

$\Theta$ is the least ordinal $\alpha$ with the property that there is no surjection $f:\mathbb {R}\rightarrow \alpha$. ${\mathsf {AD}}_{\mathbb {R}}$ is the Axiom of Determinacy for games played on the reals. It asserts that every game of length $\omega$ of perfect information in which players take turns to play reals is determined. An ideal $\mathcal {I}$ on $\omega _1$ is $\omega _1$-dense if the boolean algebra ${\wp }(\omega _1)/ \mathcal {I}$ has a dense subset of size $\omega _1$. We consider the theories, where $\mathsf {CH}$ stands for the Continuum Hypothesis, \begin{gather*} \mathsf {ZFC} + \mathsf {CH} + \text {“There is an $\omega _1$-dense ideal on $\omega _1$.''}\\ \mathsf {ZF}+{\mathsf {AD}}_{\mathbb {R}} + \text {“$\Theta $ is a regular cardinal.''}\end{gather*} The main result of this paper is that the first theory given above implies the existence of a class model of the second theory given above. Woodin, in unpublished work, showed that the consistency of the second equation given above implies the consistency of the first equation given above. We will also give a proof of this result, which, together with our main theorem, establish the equiconsistency of both the equations given above.

As a consequence, this resolves part of question 12 of W. Hugh Woodin [The axiom of determinacy, forcing axioms, and the nonstationary ideal, Walter de Gruyter & Co., Berlin, 1999], in particular, it shows that the theories (b) and (c) in question 12 of W. Hugh Woodin [The axiom of determinacy, forcing axioms, and the nonstationary ideal, Walter de Gruyter & Co., Berlin, 1999] are equiconsistent. Thus, our work completes the work that was started by Woodin and Ketchersid in [Toward AD($\mathbb {R}$) from the continuum hypothesis and an $\omega _1$-dense ideal, ProQuest LLC, Ann Arbor, MI, 2000] some 25 years ago. We also establish other theorems of similar nature in this paper, showing the equiconsistency of the second equation given above and the statement that the non-stationary ideal on ${\wp }_{\omega _1}(\mathbb {R})$ is strong and pseudo-homogeneous. The aforementioned results are the only known equiconsistency results at the level of $\mathsf {AD}_{\mathbb {R}} + \text {“$Θ$ is a regular cardinal.''}$