The omega-one-strongly compact refinement and open gap

Bagaria and da Silva prove that $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C}+\text{there is an }{\omega }_1\text{-strongly compact cardinal})$ implies $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{N}\mathrm{M}\mathrm{S}\mathrm{C})$: a random-real extension makes PMEA-$\sigma$ (PMEA and PMEA-sigma) hold, and PMEA-$\sigma$ already implies the normal Moore space conjecture (PMEA implies the normal Moore space conjecture). This lowers the large cardinal in the consistency upper bound from strongly compact to ${\omega }_1$-strongly compact. Whether NMSC conversely implies (the consistency of) an ${\omega }_1$-strongly compact cardinal is recorded there as an open question, and is not asserted or refuted here.

Status of the claim in this library. This is orientation only. The pullback computation of Theorem 2.10 is given there with its Solovay-measure input sketched and the PMEA-$\sigma$ separation modification omitted, so the item is not used as a proof supplier and carries no proof load; the proved consistency upper bound used on this page is the strongly compact one from the random algebra interface (Strong compactness and the product-measure extension interface).

• What is and is not claimed. Only the relative-consistency implication is recorded; no ${\omega }_1$-strongly compact cardinal is asserted to exist, and no implication between NMSC and ${\omega }_1$-strong compactness is claimed in either direction.