Topological applications of generic huge embeddings

Abstract

In the Foreman-Laver model obtained by huge cardinal collapse, for many $\Phi ,\Phi ({\aleph _1})$ implies $\Phi ({\aleph _2})$. There are a variety of set-theoretic and topological applications, in particular to paracompactness. The key tools are generic huge embeddings and preservation via $\kappa$-centred forcing. We also formulate "potent axioms" à la Foreman which enable us to transfer from ${\aleph _1}$ to all cardinals. One such axiom implies that all ${\aleph _1}$-collectionwise normal Moore spaces are metrizable. It also implies (as does Martin’s Maximum) that a first countable generalized ordered space is hereditarily paracompact iff every subspace of size ${\aleph _1}$ is paracompact.