On Harrington's model in which Separation holds but Reduction fails at the 3rd projective level, and on some related models of Sami

TL;DR

This paper discusses a 1975 handwritten note by Harrington that constructs a ZFC model where the Pi-1-3 separation principle holds but the Sigma-1-3 reduction principle fails, highlighting a nuanced aspect of descriptive set theory.

Contribution

It provides a detailed analysis of Harrington's original construction, illustrating a specific separation between separation and reduction principles at the third projective level.

Findings

Harrington's model demonstrates separation without reduction at the third projective level.

The construction clarifies the relationship between separation and reduction in descriptive set theory.

The paper connects Harrington's work to related models of Sami.

Abstract

In a handwtitten note of 1975, Leo Harrington sketched a construction of a model of ZFC (no large cardinals or anything beyond ZFC!) in which $\mathbf\Pi^1_3$-Separation holds but $\mathbf\Sigma^1_3$-Reduction fails. The result has never appeared in a journal or book publication except for a few of old references.