Suitable extender models. II: Beyond \(\omega \)-huge (Q2883843)

This journal-filling paper (more than 300 pages), referred to as Part II, is the continuation of [the author, ``Suitable extender models. I'', ibid. 10, No. 1--2, 101--339 (2010; Zbl 1247.03110)], referred to as Part I. The author urges the reader who undertakes a detailed reading of Part II to have Part I available for consultation, as Part II makes many references to its contents. Also, the introduction to Part I presents an overview of both papers.NEWLINENEWLINEPart II: The author considers large-cardinal axioms at the level of \(\omega\)-huge. The original impetus for his study was to understand how these axioms transfer down to the suitable extender models of Part I. The investigation of the transference problem unearths an analog of \(\text{AD}_{\mathbb R}\) at the level of \(\omega\)-huge. The author shows that the resulting \(\text{AD}_{\mathbb R}\)-like axiom transfers from \(V\) to suitable extender models. In addition, the transference proof reveals that there is a proper class of \(\lambda\) at which the \(\text{AD}_{\mathbb R}\)-like axioms hold and the existence of this class is invariant under set forcing.NEWLINENEWLINEThe discovery of \(\text{AD}_{\mathbb R}\)-like axioms motivates the author to propose two conjectures: (1) the weak uniqueness of square roots at \(\lambda\), and (2) the AD-conjecture at \(\lambda\) (the subjects of Section 5 and Sections 6 and 8, respectively). For \(\mathbb M\) a suitable extender model, the validity of either conjecture would show that all such determinacy-like axioms (ranked by a suitable ordinal parameter) transfer from \(V\) to a small generic extension of \(\mathbb M\) and that a considerable initial segment of them transfer to \(\mathbb M\).NEWLINENEWLINENEWLINENEWLINE[For part I see ibid. 10, No.\,2, 101--339 (2011; Zbl 1247.03110).]