Determinacy and extended sharp functions on the reals. II: Obtaining sharps from determinacy
This paper explores the connection between determinacy and large cardinal-like axioms called \(\#\)'s (sharps). The first to connect large cardinals and determinacy was Solovay, who showed that the axiom of determinacy (AD) implies that \(\omega_ 1\) is a measurable cardinal. Martin showed that existence of measurable cardinals implies the determinacy of \(\Pi^ 1_ 1\)-games. His proof, in fact, only used the weaker hypothesis that \(0^ \#\) exists. Building on work of Friedman, Martin, and Solovay, Harrington showed \(0^ \#\) implies that \(\Pi^ 1_ 1\) games are determined. These authors had also shown that \(L_ \mu\) (the minimal model for a measurable cardinal) follows from the determinacy of \((\omega^ 2+1)-\Pi^ 1_ 1\). Conversely, Martin has shown that the existence of \(L_ \mu\) implies \(\Delta((\omega^ 2+1)- \Pi^ 1_ 1)\) games are determined. The class \((\omega^ 2+1)-\Pi^ 1_ 1\) refers to \(\omega^ 2+1\) level of the difference hierarchy. It is a small subclass of the \(\Delta^ 1_ 2\) sets, in fact it is a small subclass of the \(\sigma\)-algebra generated by the \(\Pi^ 1_ 1\) sets. This paper (with part I [ibid. 55, No. 3, 237-263 (1992; Zbl 0745.03042)]) establishes the equivalence of \((\beta+1)\#^ 1_{\gamma+1}(0)\) exists and determinacy of \((\gamma *\Pi^ 0_ 1,\beta *\Sigma^ 0_ 1)_ +^*\) games.
scientific article; zbMATH DE number 64028
scientific article