Set theory ZFC- is ZFC without power set, but with replacement. It does not imply the collection axiom scheme, as discussed in <http://jdh.hamkins.org/what-is-the-theory-zfc-without-power-set/>


Does consistency of ZFC- imply consistency of that plus collection? I do not mind taking all of ZFC- as metatheory.



Consistency of the collection axiom scheme compared to replacement
The answer is yes, because you can go to $L$.


More specifically, if $M$ is a model of ZFC-, defined with
replacement, then consider $L^M$, the class of sets that $M$
thinks arise in the constructible hierarchy. It is not difficult
to see that $M$ thinks $L\_\alpha$ exists for every ordinal
$\alpha$ that it has, and ultimately one can see that $L^M$ is a
model of ZFC-, since instances of replacement inside $L^M$ amount
to instances of replacement in $M$, since $L^M$ is definable in
$M$. But now, since $L^M$ has a definable well-ordering of the
universe, it has a definable Skolem function and this means that
$L^M$ models $\text{ZFC}^-$, defined with collection+separation.


So the consistency of ZFC-, defined with replacement, implies the
consistency of $\text{ZFC}^-$, defined with collection+separation.


That said, the main point of our article is that ZFC-, defined
with replacement, is the wrong version of this theory, since so
many things go wrong in it.


