Paper detail

Template iterations with non-definable ccc forcing notions

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 $κ$ is a measurable cardinal and $θ<κ<μ<λ$ are uncountable regular cardinals, then there is a ccc poset forcing $\mathfrak{s}=θ<\mathfrak{b}=μ<\mathfrak{a}=λ$. Another application is to get models with large continuum where the groupwise-density number $\mathfrak{g}$ assumes an arbitrary regular value.