[EDIT: The axiom of successor cardinals was found by an answer by Greg Kirmayer, not to be capturing the intended meaning of it, which is simply reflected by its name, i.e. the existence of a successor cardinal for every cardinal. A corrective note had been inserted below that axiom.

Is Z + Rank + Successor cardinals + Ordinal inaccessibility = ZF?

Where:

**Ranks:** $\forall x \exists \alpha: x \in V_\alpha$

where $V_\alpha$ is the $\alpha^{th}$ stage of Von Neumann's universe (the cumulative hierarchy).

**Successor cardinals:** if $f$ is a definable function, then: $$\forall \text{ordinal } \alpha \ \exists \text{ ordinal }\beta \, (\neg \forall \gamma < \beta \exists \lambda < \alpha : f(\lambda)=\gamma)$$ By $\text{ordinal}$ it means a *von Neumann* ordinal defined in the usual manner.

For every ordinal there is an ordinal such that no surjective function (definable in the language of set theory) from the former to the latter can exist.

[NOTE]: The above formulation suffers a flaw, therefore the correct formulation is: $$\forall \text{ordinal } \alpha \, \exists \text{ordinal } \beta: \not \exists f (f: \beta \hookrightarrow \alpha)$$, where $\hookrightarrow$ signify "injective function".

**Ordinal inaccessibility:** if $f$ is a definable function, then: $$ \forall \gamma \, (\neg \forall \text{ ordinal } \alpha \,\exists \beta < \gamma : \alpha \leq f(\beta))$$ Any function [definable in the language of set theory] coming from an ordinal cannot have every ordinal being smaller than or equal to some ordinal in its range. That is, its range cannot be a cofinal subclass of the class of all ordinals.

The idea is that every uncountable cardinal in ZF is either Regular and therefore a successor cardinal, or otherwise a singular limit cardinal. Here, both of those kinds would be constructed from below, and the axiom of Rank assures that all sets are built successively within those ordinally indexed stages. Ordinal inaccessibility is I think equivalent to *Ordinal Replacement* which by itself is actually weak, it can only build stages up to $V_{\omega_1}$, and of course successor cardinals can only build the next stages, but together it seems that they can act to build up the whole of Von Neumann's universe. So, I thought that the above would prove full Replacement. Its easy to prove the result with Choice (in the form of every set is bijective to some von Neumann ordinal); but without it the proof is elluding me?

ordinalin this theory? And I think there is a typo in the definition of ordinal inaccessibility; is $\lambda=\gamma$? $\endgroup$strictly well orderedby $\in$. An this is the usual official definition of von Neumann ordinals, and it is also equivalent to the one I gave in my prior comment. $\endgroup$founded, but not in general $\in$-wellordered. $\endgroup$2more comments