Measuring club-sequences with a large continuum

TL;DR

This paper addresses the challenge of constructing models with large continuum sizes in set theory, focusing on measuring, a strong negation of Club Guessing, and provides new techniques for this purpose.

Contribution

It introduces novel methods for producing models with continuum larger than the second uncountable cardinal in the context of measuring.

Findings

Developed techniques for models with large continuum

Extended understanding of measuring in set theory

Provided solutions to longstanding problems in iterated proper forcing

Abstract

One of the most frustrating problems faced by set theorists working with iterated proper forcing is the lack of techniques for producing models in which the continuum has size greater than the second uncountable cardinal. In this paper we solve this problem in the specific case of measuring, a very strong negation of Club Guessing introduced by Justin Moore.