Within ZFC, the [von Neumann hierarchy](https://en.wikipedia.org/wiki/Von_Neumann_universe) consists of sets $V\_\alpha$ indexed by ordinals, subject to the following conditions:


* $V\_0=\varnothing$.
* $V\_{\alpha+1}=\mathcal P(V\_\alpha)$.
* $V\_\lambda=\bigcup\_{\beta<\lambda}V\_\beta$ for limit $\lambda$.


My question is: **what is the formal justification for the last step?**


The [axiom of union](https://en.wikipedia.org/wiki/Axiom_of_union) would allow us to construct $V\_\lambda$ if we already had a set $\{V\_\beta:\beta<\lambda \}$. However, the existence of $V\_\beta$ for $\beta<\lambda$ does not obviously guarantee the existence of this set: $V\_{\omega\cdot 2}$ in ZF famously models itself minus the [axiom of replacement](https://en.wikipedia.org/wiki/Axiom_schema_of_replacement). This of course suggests that with replacement, this set could be constructed as the image of $\lambda$ (which under the von Neumann ordinal construction is the set of all lower ordinals) under the class function $V$.


But then $V$ would have to be a class function somehow defined through transfinite induction. Since predicates can't refer to themselves, and since we can't just assert the existence and uniqueness of a class function as a theorem the way we can with normal functions, the way this works eludes me.



Replacement axiom and the von Neumann hierarchy
This is really just a long comment, but the phrasing on the wikipedia page for replacement is verbose and perhaps obscuring how to use it here. Consider this version of replacement:



> 
> For any set $X$ and binary predicate $\phi(-,-)$ such that for each element $x\in X$ there exists a unique set $y\_x$ such that $\phi(x,y\_x)$ is true, there exists a set $Y$ whose members are precisely the sets $y\_x$ such that there exists some $x\in X$ with $\phi(x,y\_x)$ true. We denote the set $Y$ guaranteed by this axiom together with a set $X$ and binary predicate $\phi(-,-)$ by $$\{y\_x:x\in X\}.$$
> 
> 
> 


This is equivalent to all other standard phrasings of replacement over the rest of the $ZFC$ axioms, and is easier to use in the situation you outiline above. Specifically, take $X=\lambda$ and let $$\phi(-,-)=\text{All sets up to rank $-$ are members of $-$, and nothing else.}$$ which accepts ordinals in the first argument and arbitrary sets in the second. For each $\beta<\lambda$ we have that $\phi(\beta,V\_\beta)$ is true, and if $\phi(\beta,Z)$ is true for some other set $Z$ then $Z=V\_\beta$ by extensionality, so $V\_\beta$ is unique satisfying $\phi(\beta,V\_\beta)$ for all $\beta<\lambda$. Consequently $$\{V\_\beta:\beta\in\lambda\}$$ is a set by replacement, and the union of this set is $V\_\lambda$.


