Recall that Easton forcing was introduced to show that the continuum function at regular cardinals could be anything subject to 'the obvious constraints' (monotonicity etc). However, it is a handy method if one wants to add a proper class of sets. My question is why
do would we now restrict to using forcing conditions only using regular cardinals (edit: if we wanted only to add class-many sets)? I've had a read through Friedman's Class forcing, and all the (nontrivial) examples given there are variants on Easton forcing, only playing with things like supports and stationarity.
I'm not interested in preserving AC, though I suspect that we lose tameness at some point, and hence axioms like powerset may fail to hold.
Hmm, let me state my actual question, which was in fact rather implicit (and everyone's comments/answers have helped me figure out how to phrase it, so thank you all).
If I try to add $F(\kappa)$ generic subsets to each cardinal $\kappa$ (by some simple class function $\kappa \mapsto F(\kappa)$, such as the identity, or constant at some given infinite cardinal), will I get a model of ZF(C)? Or is the restriction to adding subsets to only regular cardinals, as in Easton forcing, necessary?