A strong antidiamond principle compatible with CH

TL;DR

This paper demonstrates the consistency of a strong antidiamond principle with the Continuum Hypothesis by developing a novel forcing construction using Shelah's properness theory and definability techniques.

Contribution

It introduces a new method for constructing properness parameters alongside NNR iterations, extending Shelah's theory with definability in third order arithmetic.

Findings

Proves the consistency of the strong antidiamond principle with CH.

Develops a new technique for handling properness parameters in forcing iterations.

Uses definability in third order arithmetic to manage complex forcing notions.

Abstract

A strong antidiamond principle (*c) is shown to be consistent with CH. This principle can be stated as a "P-ideal dichotomy": every P-ideal on omega-1 (i.e. an ideal that is sigma-directed under inclusion modulo finite) either has a closed unbounded subset of omega-1 locally inside of it, or else has a stationary subset of omega-1 orthogonal to it. We rely on Shelah's theory of parameterized properness for NNR iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application of the NNR iteration theory involves definability of forcing notions in third order arithmetic, analogous to Souslin forcing in second order arithmetic.