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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.

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    $\begingroup$ You asked and deleted this question a while ago. $\endgroup$
    – LSpice
    Oct 11 '20 at 0:38
  • $\begingroup$ The question has been changed (completely). $\endgroup$
    – hermes
    Oct 11 '20 at 15:41
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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.

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  • $\begingroup$ 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 Burali-Forti 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 '20 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 '20 at 20:10

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