Bourbaki--Zorn Normal Forms for Maximality Arguments

Abstract: We isolate a normal-form mechanism underlying the Bourbaki--Witt fixed-point theorem and least-upper-bound versions of Zorn-type maximality principles. Given a progressive self-map on a partially ordered set, we define a Bourbaki tower as a well-ordered trajectory closed under successor steps and least upper bounds of initial segments. Under the hypothesis that every well-ordered subset admits a least upper bound, every progressive self-map generates a largest Bourbaki tower, whose terminal least upper bound is a fixed point. Consequently, strictly progressive self-maps cannot exist in such posets. Combining this obstruction with a choice selector on strict upper cones yields a concise maximality principle: if every well-ordered subset has a least upper bound, then the poset has a maximal element. The contribution is methodological rather than axiomatic; the paper makes explicit a reusable proof architecture connecting Bourbaki--Witt fixed points, strict progression obstructions, and least-upper-bound versions of Zorn-type maximality arguments.