If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ-strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to cons...

Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V . In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.

We show that it is consistent, relative to ω many supercompact cardinals, that the super tree property holds at אn for all 2 ≤ n < ω but there are weak square and a very good scale at אω .

To a large extent, the scientific work of both Peter Koepke and Philip Welch has always been inspired by Ronald Jensen’s fine structure theory and his Covering Lemma for L, the constructible universe. Cf., e.g., their joint papers [5] and [6]. In this paper we aim to play with the theme that large cardinals compatible with “V=L,” specifically: remarkable cardinals, allow us to create situations...

We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ZF + ¬ACω + Every successor cardinal is regular + E...