Large continuum, oracles

TL;DR

This paper develops a method using iterated forcing with oracles to enlarge the continuum beyond aleph_2, allowing precise control over cardinal characteristics like the continuum and covering number of meager sets.

Contribution

It generalizes previous work by replacing alephs with larger cardinals and introduces a new approach using absolute c.c.c. conditions in iterated forcing.

Findings

The continuum can be made larger than aleph_2.

The method achieves specific values for cov(meagre).

It provides applications for controlling cardinal characteristics.

Abstract

Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting with lambda=lambda^{<lambda}>aleph_1). Well, instead of properness we demand absolute c.c.c. So we get, e.g. the continuum is lambda^+ but we can get cov(meagre)=lambda. We give some applications. As in math.LO/0303294, it is a "partial" countable support iteration but it is c.c.c.