In the von Neumann universe (also known as the cumulative hierarchy), the rank $R(x)$ is defined as the least ordinal $\alpha$ that $x\in V_{\alpha +1}$ (or equivalently $x\subset V_{\alpha}$). I'd like to know who gave this definition and when.

6$\begingroup$ You asked and deleted this question a while ago. $\endgroup$ – LSpice Oct 11 at 0:38

$\begingroup$ The question has been changed (completely). $\endgroup$ – hermes Oct 11 at 15:41
The modern definition of rank appears to have arisen gradually. The introduction of Christine Knoche's $1973$ masters thesis gives a good summary: it seems to have begun with Mirimanoff in $1917$ and been given its modern form by Tarski in $1955$. Along the way von Neumann, Russell, and Bernays (and others) played with it in various ways.
I think it would ultimately be an oversimplification to try to credit it to any individual author  nor is there any particular need to.

$\begingroup$ I do not think it should be credited to Mirimanoff because he only raised a vague idea of rank in his seminal paper "Les antinomies de Russell et de BuraliForti et le probleme fondamental de la theorie des ensembles" in 1917, while the von Neumann universe came after 1930 (in Zermelo's paper). $\endgroup$ – hermes Oct 11 at 15:58

$\begingroup$ @ Noah Schweber, thanks a lot. Yes, knoche's thesis provides a good summary on rank function upon which a lot of research has been done already. $\endgroup$ – hermes Oct 11 at 20:10