We say that a regular cardinal κ,, has the tree property if there are no κ‐Aronszajn trees; we say that κ has the weak tree property if there are no special κ‐Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal,, is consistent with an arbitrary continuum function below which satisfies,. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal,, is consistent with an (...) arbitrary continuum function below which satisfies,. Thus the tree property has no provable effect on the continuum function below except for the trivial requirement that the tree property at implies for every infinite κ. (shrink)

We show that if \ is \\)-hypermeasurable for some cardinal \ with \ \le \mu \) and GCH holds, then we can extend the universe by a cofinality-preserving forcing to obtain a model \ in which the \\)-hypermeasurability of \ is indestructible by the Cohen forcing at \ of any length up to \ is \\)-hypermeasurable in \). The preservation of hypermeasurability is useful for subsequent arguments. The construction of \ is based on the ideas of Woodin and Cummings :1–39, (...) 1992) for preservation of measurability, but suitably generalised and simplified to achieve a more general result. Unlike the Laver preparation :385–388, 1978) for a supercompact cardinal, our preparation non-trivially increases the value of \, which is equal to \ in \ is still true in \ if we start with GCH). (shrink)

We give an application of the extender based Radin forcing to cardinal arithmetic. Assuming κ is a large enough cardinal we construct a model satisfying 2κ = κ⁺ⁿ together with 2λ = λ⁺ⁿ for each cardinal λ < κ, where 0 < n < ω. The cofinality of κ can be set arbitrarily or κ can remain inaccessible. When κ remains an inaccessible, Vκ is a model of ZFC satisfying 2λ = λ+n for all cardinals λ.