Simultaneously vanishing higher derived limits

In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

Mad families, splitting families and large continuum

Let κ < λ be regular uncountable cardinals. Using a finite support iteration (in fact a matrix iteration) of ccc posets we obtain the consistency of . If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of .

Suslin forcing and parametrized ◊ principles

By using finite support iteration of Suslin c.c.c forcing notions we construct several models which satisfy some ◊-like principles while other cardinal invariants are larger than ω1.

Finite support iteration and strong measure zero sets

Any finite support iteration of posets with precalibre ℵ1 which has the length of cofinahty greater than ω1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ1 strong measure zero sets.