ZF+DC+AX$_4$
  We consider mainly the following version of set theory:"ZF + DC and for every $\lambda,\lambda^{\aleph_0}$ is well ordered", our thesis is that this is a reasonable set theory, e.g. much can be said. In particular, we prove that for a sequence $\bar\delta = \langle \delta_s:s \in Y\rangle,\textrm{cf}(\delta_s)$ large enough compared to $Y$, we can prove the pcf theorem with minor changes (using true cofinalities not the pseudo one). We then deduce the existence of covering numbers and define and prove existence of truly successor cardinals. From this we show that some diagonalization arguments (more specifically some black boxes and consequence) on Abelian groups. We end by showing some such consequences hold in ZF above. 
Saharon Shelah
math.LO