Adding Abraham clubs and $\alpha$-properness

TL;DR

This paper introduces a new variant of Abraham forcing that adds a club in , which is proper for all smaller indecomposable ordinals but not for the ordinal itself, expanding the understanding of properness in forcing.

Contribution

The paper develops a novel Abraham forcing variant that distinguishes between different levels of properness based on indecomposable ordinals, specifically introducing -appropriate clubs.

Findings

The new forcing is <-proper but not -proper.

It demonstrates a hierarchy of properness related to indecomposable ordinals.

Provides a framework for analyzing club-adding forcings with ordinal-specific properness.

Abstract

For every indecomposable ordinal $\alpha < \omega_1$, we introduce a variant of Abraham forcing for adding a club in $\omega_1$, which is $<\alpha$-proper but not $\alpha$-proper.