By "Parameter free Zermelo" I mean a theory defined in the language of set theory that has exactly Zermelo set theory axioms except the axiom scheme of Separation which is replaced by parameter free separation scheme, the later is formally written as:


Parameter free separation scheme: if $\phi(y)$ is a formula in which only the symbol "$y$" occur free, then:


$\forall A \space \exists x \space \forall y \space (y \in x \iff y \in A \wedge \phi(y))$


is an axiom.


My question: is there a known result about provability of Cantor's theorem (that a set is strictly smaller than its power) in this parameter free Zermelo?


I personally think it is not provable and that Parameter free Zermelo is consistent with the axiom that all sets are countable? I tried to search for that result but I didn't see it mentioned anywhere. 



Can Cantor's theorem be proved in Parameter Free Zermelo?
There is an answer to this question of mine in the following article that was refered to here by Goldstern, see theorem 0.7. Although the author stated it in $ZC^o$, but the argument makes no use of Choice whatsoever. Here is the link:


<http://wwwmath.uni-muenster.de/u/rds/ZFC_without_parameters.pdf>


The reason why such a proof was possible relied on a version of Zermelo that can define pairs of arbitrary sets and also defines set union of any set and of course power set of any set without resorting to Separation in the first place. If we take Zermelo to be the theory axiomatized by the axioms of ZF presented in the Wikipedia at:


<https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory>


taking out from them the Schema of Replament and Axiom of Choice. Then clearly we'll be needing full spearation (i.e. with parameters) in order to define pairs, unions and powers. So if by Parameter free Zermelo it is meant Zermelo so axiomatized but with replacing Separation by a parameter free version of it, then it appears to me that the above proof would be blocked. And I think David Libert's construction provides a counter-example to it.


