Take the 2-minute tour ×  Hello all, one may look for "minimal
system of axioms" for ZFC (or any other theory) in the following
(unusual) sense : say that a subset S of ZFC is "sufficient" if
there is an explicit procedure that constructs a model of ZFC
from any model of S.  Thus, for example, ZF is sufficient since
inside ZF we can construct Godel's universe L which is a model
for ZFC. My questions : are minimal sufficient subsets of ZFC
known? Is extensionality+infinity+(power set)+(separation
scheme) sufficient?  share|improve this question  5 Answers 5  I
think that you haven't quite asked the question you intended to
ask, since $L$ is not a set, but rather a proper class, and of
course by the incompleteness theorem no subtheory of ZFC, if
consistent, can prove that there is a set model of ZFC. So your
remarks about ZF having the property you state are not correct.
Rather, it seems that what you want is a minimal or very weak
subtheory $S$ of ZFC such that from any model of $S$ we can
construct a model o