I am talking about the principle that says that every set is the image of a projective 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 (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 ?

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