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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6$\begingroup$ You asked and deleted this question a while ago. $\endgroup$– LSpiceCommented Oct 11, 2020 at 0:38
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$\begingroup$ The question has been changed (completely). $\endgroup$– Eugene ZhangCommented Oct 11, 2020 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.
answered Oct 11, 2020 at 0:55
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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). $\endgroup$ Commented Oct 11, 2020 at 15:58
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$\begingroup$ @ 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. $\endgroup$ Commented Oct 11, 2020 at 20:10
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$\begingroup$ @EugeneZhang: I encourage you to read Mirimanoff's paper in French or my annotated English translation and tell me your comments by private email. The rank is clearly defined in paragraph 7. He also anticipates the representation of ordinals and set-theoretic hierarchy that were later called after von Neumann, as well as the axiom-scheme of replacement. This was the first of his three papers on set theory, most of his other work being on number theory. $\endgroup$ Commented Oct 14 at 14:48
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$\begingroup$ @Paul Taylor, thanks for your translation. I donot believe Mirimanoff gives the current definition of rank clearly. However, it is not important because this rank is wrong. The correct rank should be the least ordinal $α$ that $x∈V_α$. See my paper at arxiv.org/abs/2304.00581 (section 1). $\endgroup$ Commented Oct 14 at 17:59
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$\begingroup$ @EugeneZhang: I would consider Mirimanoff's definition to be correct; there should be a Lemma to say that yours is equivalent. (In own research, I would prefer to say that the rank is the categorical refection of sets into ordinals, ie the universal imposition of transitivity.) $\endgroup$ Commented Oct 14 at 18:23