Determinacy and extended sharp functions on the reals. II: Obtaining sharps from determinacy
type: Item
instance of: scholarly article
title: Determinacy and extended sharp functions on the reals. II: Obtaining sharps from determinacy
zbMATH Open document ID: 0759.03026
DOI: 10.1016/0168-0072(92)90032-U
author: Derrick Albert DuBose
published in: Annals of Pure and Applied Logic
publication date: 1992-09-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03E60
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 64028
zbMATH Keywords: determinacy
zbMATH Keywords: sharps
MaRDI profile type: Publication
cites work: The equivalence of determinacy and iterated sharps
cites work: Determinacy and the sharp function on the reals
cites work: Determinacy and the sharp function on objects of type k
cites work: Determinateness in the low protective hierarchy
cites work: Analytic determinacy and 0#
cites work: Measurable cardinals and analytic games
cites work: Borel determinacy
cites work: An extension of Borel determinacy
cites work: Descriptive set theory
full work available at URL: 0168-0072(92)90032-u
OpenAlex ID: W2065464598
Recommended article: Determinacy and the sharp function on the reals
Recommended article: Q5489366
Recommended article: Determinacy in L(ℝ)
Recommended article: An extension of Borel determinacy
Recommended article: PROVABLY GAMES
Recommended article: Q4934562
Recommended article: Determinacy separations for class games
Recommended article: Determinateness of certain almost-Borel games
Recommended article: OPTIMAL PROOFS OF DETERMINACY II
Recommended article: THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY
description: scientific article; zbMATH DE number 64028