Non-reflection of the bad set for {check I}_theta[lambda] and pcf

TL;DR

This paper investigates reflection properties of certain ideals and sets in set theory, providing new insights into their structure and size, and exploring the existence of free black boxes in the context of pcf theory.

Contribution

It advances understanding of reflection phenomena for bad sets and sizes of related ideals, and explores the existence of free black boxes in pcf theory.

Findings

Reflection properties of bad sets are characterized under certain conditions.

Sizes of specific ideals related to increasing cofinal sequences are determined.

Existence of free black boxes in pcf theory is established or examined.

Abstract

We reconsider here the following related pcf questions and make some advances: (Q1) concerning the ideal {check I}_kappa [lambda] how much reflection do we have for the bad set S^{bd}_{lambda, kappa} subseteq {delta < lambda : cf(delta)= kappa} assuming it is well defined? (Q2) for an ideal J on kappa how large are S^{bd}_J[f],S^{ch}_J[f] for f=< f_alpha : alpha<lambda > which is <_J-increasing and cofinal in (prod limits_{i< kappa} lambda_i,<_J) ? (Q3) are there somewhat free black boxes?