Starting from cardinals κ κ is measurable, we construct a model for the theory “ZF + n < ω[DCn] + ω + 1 is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of ω + 1, and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possible.

We obtain 2 models in which AC is false and in which there are long sequences of consecutive large cardinals.

Given sets and a regular cardinal μ, let be the statement that for any function, there are functions and such that for all,. In, the statement is false. However, we show the theory (which is implied by + “” + “ω1 is measurable”) implies that for every there is a such that in some inner model, κ is measurable with Mitchell order.