Instances of dependent choice and the measurability of $\text{ℵ}{}_{\mathrm{\omega \; +\; 1}}$

Abstract

Starting from cardinals κ < λ where κ is 2^λ supercompact and λ > κ is measurable, we construct a model for the theory “ is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of $\text{ℵ}{}_{\mathrm{\omega \; +\; 1}}$, and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possible.