We consider mainly the following version of set theory: "ZF+DC and for every $${\lambda, \lambda^{\aleph_0}}$$?,??0 is well ordered", our thesis is that this is a reasonable set theory, e.g. on the one hand it is much weaker than full choice, and on the other hand much can be said or at least this is what the present work tries to indicate. In particular, we prove that for a sequence $${\overline{\delta} = \langle\delta_{s}: s \in Y\rangle, {\rm cf}(\delta_{s})}$$?¯=??s:s?Y?,cf(?s) large enough compared to Y, we can prove the pcf theorem with minor changes (in particular, using true cofinalities not the pseudo ones).We then deduce the existence of covering numbers and define and prove existence of a class of true successor cardinals. Using this we give some diagonalization arguments (more specifically some black boxes and consequences) on Abelian groups, chosen as a characteristic case.We end by showing that some such consequences hold even in ZF above.
['Saharon Shelah']
ZF + DC + AX4