The Ramified Analytical Hierarchy using Extended Logics

TL;DR

This paper explores how extended logics can replace second-order definability in the Ramified Analytical Hierarchy, enabling the construction of minimal models of analysis with varied definability properties.

Contribution

It introduces a method to use extended logics, including game quantifiers, to generate minimal correct models of analysis within the Ramified Hierarchical framework.

Findings

Extended logics allow defining models that differ from the original hierarchy.

Game quantifiers help obtain minimal correct models of analysis.

Models can be generated from abstract definability notions like Spector Classes.

Abstract

The use of Extended Logics to replace ordinary second order definability in Kleene's {\em Ramified Analytical Hierarchy} is investigated. This mirrors a similar investigation of Kennedy, Magidor and V\"a\"an\"anen \cite{KeMaVa2016} where G\"odel's universe $L$ of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain {\em minimal correct} models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability.