  Consider a variant of the usual story about the iterative conception of sets.
As usual, at every stage, you find all the (bland) sets of objects which you
found earlier. But you also find the result of tapping any earlier-found object
with any magic wand (from a given stock of magic wands).
  By varying the number and behaviour of the wands, we can flesh out this idea
in many different ways. This paper's main Theorem is that any loosely
constructive way of fleshing out this idea is synonymous with a ZF-like theory.
  This Theorem has rich applications; it realizes John Conway's (1976)
Mathematicians' Liberation Movement; and it connects with a lovely idea due to
Alonzo Church (1974).
