# Is full Replacement provable in Z + Ordinal Replacement?

$$\text{Ordinal Replacement:}$$ if $$\phi(x,y)$$ is a formula in two free variables $$x,y$$, then:

$$\forall x \ [ordinal(x) \to \exists! y \ (ordinal (y) \wedge \phi(x,y)) ] \to \forall A \ (\forall x \in A (ordinal(x)) \to \exists B \ \forall y \ (y \in B \leftrightarrow \exists x \in A \ \phi(x,y)))$$

is an axiom.

In English for every set $$A$$ of von Neumann ordinals and a function $$F$$ from von Neumann ordinals to von Neumann ordinals, then we can get a set $$B$$ whose elements are all the $$F$$-images of elements of $$A$$.

i.e. we can replace elements of a set of ordinal by ordinals.

Now is full Replacement provable in $$\text{Z + Ordinal Replacement}$$?

No. Consider the model $\langle V_{\omega_1},\in\rangle$. This is a model of Zermelo's theory, but all ordinals in it are countable. Thus, it satisfies the ordinal-replacement axiom, since the image of a countable ordinal under any function will be countable, and $V_{\omega_1}$ contains all its countable subsets, since $\omega_1$ is regular.
In fact, in this case, we needn't restrict $B$ to consist of ordinals. Rather, we get the general version of replacement for functions whose domain is a set contained in the ordinals.