Set theory without choice: not everything on cofinality is possible

Abstract.

We prove in ZF+DC, e.g. that: if \(\mu=|{\cal H}(\mu)|\) and \(\mu>\cf(\mu)>\aleph_0\) then \(\mu ^+\) is regular but non measurable. This is in contrast with the results on measurability for \(\mu=\aleph_\omega\) due to Apter and Magidor [ApMg].