An Overview Over Least Fixed Points in Weak Set Theories

Explore the intricacies of least fixed points in weak set theories in this 27-minute lecture by Silvia Steila from the Hausdorff Center for Mathematics. Delve into the Tarski Knaster Theorem, which states that every monotone function on a complete lattice has a least fixed point. Examine two standard proofs of this theorem: one utilizing closed sets under a monotone function, and another based on inductive definition by stages. Compare how these constructions differ when applied to conservative second-order extensions of Kripke Platek Set Theory (KP), where the powerset axiom is avoided. Gain insights into the challenges of proving Tarski Knaster Theorem for the powerset of natural numbers in weaker set theories, and understand the implications for mathematical foundations and logic.