# A History of Level Theories: From Ranks to Levels.

## Davide Sutto

The paper presents a comprehensive history of the formalizations of the Iterative Conception of Set. First, I show how the core idea originated, even before Zermelo’s \(V_\alpha\)s, in reflections on ranks appearing in Mirimanoff’s 1910s contributions and earlier notes by German mathematicians. Second, I emphasize the seminal role played by Tarski’s School in developing the notion of rank and further argue that the actual genesis of the conception is to be found in Tarski’s 1950s set theory seminars, later developed in [2] and [3]. Third, I focus on the crucial passage from the notion of rank to its more explicit characterization: levels. In the work of Tarski’s School we can find the first formulations of axiomatic theories of levels in purely set theoretic terms. This is relevant as I move forward to Shoenfield and Boolos, who rather outline stage theories, adopting stages as a further primitive. Finally, I close with the more recent contributions of Potter and Button, who revitalized the notion of level by making it more tractable.

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