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.
Categories: math.LO