I am talking about the principle that says that every set is the image of a [projective](http://planetmath.org/?op=getobj&from=objects&id=6437) set. For every set $x$ there is a surjection $f:y \twoheadrightarrow x$, such that for any set $u$ and function $j:y \to u$ and surjection $g:v \twoheadrightarrow u$ there is a function $h:y \to v$ such that $j=gh$.


There is a simple model of CZF in which this holds and the axiom of choice is violated. This also violates classical logic so is not a model of ZF (= CZF + classical logic). I am curious about the following comment of Andreas Blass on ZF:



> 
> ...all the usual independence results involving weak axioms of choice remain true in
>  the presence of the negation of EPSets. In particular, none of the axioms in the 
>  appendix of [Jech, the Axiom of Choice], except AC itself, implies EPSets. 
>  The following result ... is essentially all that is known about the strength of EPSets...
> 
> 
> Theorem 6.2 EPSets implies the axiom of dependent choice ...
> 
> 
> In particular, we do not know whether it implies AC ... we conjecture that it does not.
> 
> 
> 


(Injectivity, Projectivity, and the Axiom of Choice, 1979)


[Consequence of the Axiom of Choice](http://consequences.emich.edu/CONSEQ.HTM) (1998) listed the question of whether it implies AC as still open, and in fact unless I missed something it listed no new non-trivial results.


So, in 2010, is there really still nothing to be said other than ZF+DC < ZF+EPSets $\le$ ZFC ?



Existence of enough projectives in the category of sets
I haven't heard any new information about "enough projective sets" in the classical ZF context. Your formulation, however, looks weaker. "Enough projective sets" should say that for every $x$ there is a surjection $f:y \to x$ such that every surjection from any $w$ to $y$ splits. That implies your formulation (by taking $w$ to be the pullback of $y$ and $z$ over $x$) but I don't see the converse. 


The principle "enough projective sets" has been used in constructive mathematics by Peter Aczel (and probably others) under the name "presentation axiom".


