When one reads the Wikipedia article on the Von Neumann Universe, one gets the impression that the idea of "the cumulative hierarchy" serves as a motivation for $ZFC$. I don't see really how this is the case. I don't see any of the definitions given to the cumulative hierarchy in that page implying Replacement at all.

Also when one reads in Boolos-The Iterative Conception of Set, page: 228, one gets the following:

There is an extension of the stage theory from which the axioms of replacement could have been derived. We could have taken as axioms all instances of a principle which may be put, 'If each set is correlated with at least one stage (no matter how), then for any set z there is a stage s such that for each member w of z, s is later than some stage with which w is correlated'.

This bounding or cofinality principle is an attractive further thought about the interrelation of sets and stages, but it does seem to us to be a further thought, and not one that can be said to have been meant in the rough description of the iterative conception.

It appears that what Boolos is saying is that: when we extend the rough iterative conception of set with a ranking function, then we get Replacement.

EDIT: I've mis-understood Boolos here, as Noah point in his comments and answer, Boolos was not taking about extending the iterative conception of set with a ranking function. So the rest of this post addresses the first point that I've referred to that is mentioned in the Wikipedia article. However, Boolos seems to be saying that Replacement is not related to the iterative conception of sets, and that it is an extra-thought. Which in some sense backs my argumentation that I'll present below.

I'll try here to capture the notion of building a hierarchy from below in class ambiance. So let's work in mono-sorted first order logic with identity and membership.

Define: $set(x) \iff \exists y (x \in y)$

Axioms: $ID$ axioms +

*Class axioms:*

$C_1$. **Extensionality:** $\forall a,b (\forall x (x \in a \leftrightarrow x \in b) \to a=b)$

$C_2$. **Class comprehension schema:** if $\varphi$ is a formula in which $x$ is not free,
then all closures of: $\exists x \forall y (y \in x \leftrightarrow set(y) \wedge \varphi)$ are axioms.

Define: $x=\{y|\varphi\} \iff \forall y (y \in x \leftrightarrow set(y) \wedge \varphi)$

Let $V=\{x| set(x)\}$

Define: $ R\text{ is a ranking function on V } \iff R \text{ is a function on V} \wedge \\\exists < [\text {< is a well ordering relation on range}(R) \wedge \forall x,y \in V (x \in y \to R(x) < R(y))] $

Define: $r \text{ is a rank }\iff \exists x (r=R(x))$

*Hierarchy axioms:* There exists $R$ such that:

$H_1$. **Ranking:** $R$ is a ranking function on $V$

$H_2$. **Stages:** for every rank $r$: $\forall x (\forall y \in x (R(y) \leq r) \to x \in V)$

$H_3$. **Infinity:** There exists a limit rank.

$H_4$. **Height:** $\forall x \in V (\text{well ordered}(x) \to \exists y \subset range(R) [x \text { isomorphic to } y])$

/ Theory definition finished.

Now I think this clearly captures the ranking function as built form below, which is the heart behind the philosophy of iterative conception of sets [one can easily see that the axiom $H_4$ clearly depicts this building from blow direction]. However, I don't see this reaching to the strength of $ZF$? I think it might reach to the strength of the first fixed point on the $\omega$ function of von Neumann ordinals.

If I'm correct then the addition of $Replacement$ schema to the rest of axioms of $ZF$ is better be considered as a *large cardinal axiom*, rather than being viewed as grounded in the cumulative hierarchy concept. Accordingly Replacement is to be grounded in *limitation of size* concept, that a set sized definable (parameters allowed) class is a set, and this is a notion about cardinality, rather than it being a notion about a Hierarchy or ranking or stages or iteration or the alike. A possible backing to this view is that presented by Randall Holmes here. However I'm still not sure of the above, since there is a lot of talk about the cumulative hierarchy constituting a motivation for axioms of ZFC is already well known, hence my question:

- Is replacement provable in the above ranked Hierarchy class theory?
- IF not, then how are we to understand that having the von Neumann universe constitute a motivation for Replacement schema?

notmotivated by the cumulative hierarchy per se, and I think his argument is a pretty good one. $\endgroup$ – Timothy Chow Nov 24 '18 at 5:10notReplacement?" since he thinks that Replacement is a basic feature of our conception of sets. Not every axiom has to be justified directly from the cumulative hierarchy; e.g., we don't justify Extensionality by arguing that Extensionality is motivated by the cumulative hierarchy. $\endgroup$ – Timothy Chow Nov 25 '18 at 15:411more comments