Template iterations with non-definable ccc forcing notions

TL;DR

This paper extends Shelah's iterated forcing theory to non-definable forcing notions, demonstrating its application in constructing models with specific cardinal characteristics and large continuum, including arbitrary regular values for the groupwise-density number.

Contribution

It introduces a version of Shelah's iterated forcing along a template using non-definable forcing notions, expanding the method's applicability.

Findings

Constructed a ccc poset forcing specific cardinal inequalities.

Produced models with large continuum and prescribed groupwise-density number.

Extended Shelah's theory to non-definable forcing notions.

Abstract

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are uncountable regular cardinals, then there is a ccc poset forcing $\mathfrak{s}=\theta<\mathfrak{b}=\mu<\mathfrak{a}=\lambda$. Another application is to get models with large continuum where the groupwise-density number $\mathfrak{g}$ assumes an arbitrary regular value.