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?