Generalisations of Gödel's universe of constructible sets

Gödel’s universe L of constructible sets has many attractive features. It has a definable wellordering (a strong form of AC) and satisfies not only the generalised continuum hypothesis (GCH), but also strong combinatorial principles such as Jensen’s ♦, 2 and Morass (see [6]). In this sense, the theory ZFC + V = L is mathematically strong. However many interesting set-theoretic statements imply the consistency of ZFC, whereas V = L does not. In this sense, the theory ZFC + V = L is consistency weak. For this reason it is common in set theory to assume at least the existence of inner models of V which contain large cardinals (inaccessible, measurable, strong, Woodin, superstrong and beyond). ZFC + large cardinals is consis-tency strong, in the sense that for an abundance of set-theoretic statements ϕ (not known to be inconsistent), we have Con(ZFC + LC) → Con(ZFC + ϕ) for some large cardinal axiom LC. And in many cases, we have