We generalize ∇(A), which was introduced in [Sch∞], to larger cardinals. For a regular cardinal κ>ℵ0 we denote by ∇ κ (A) the statement that and for all regular θ>κ, is stationary in It was shown in [Sch∞] that can hold in a set-generic extension of L. We here prove that can hold in a set-generic extension of L as well. In both cases we in fact get equiconsistency theorems. This strengthens results of [Rä00] and [Rä01]. is equivalent with the (...) existence of 0#. (shrink)

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s found (...) one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L. One of the nice things about AD is that the theory ZF + AD + V = L appears as a choiceless “completion” of ZF in that any interesting question seems to find an at least attractive answer in that theory. Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension. (shrink)