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