Title: The Story of 0^#

Speaker: Linfeng Wang (Peking University)

Time: 2025/4/8 (Tuesday) 15:10-18:00

Location: Room 215, Natural ​Sciences Teaching Building (理教), Peking University

Abstract:

This report explores fundamental developments in mathematical logic surrounding Gödel's constructible universe L and its relationship with large cardinal axioms. The investigation begins with Gödel's seminal work introducing constructible sets to establish the relative consistency of the Axiom of Choice and the Generalized Continuum Hypothesis. We then examine subsequent breakthroughs by Scott and Gaifman, who demonstrated that the existence of measurable cardinals contradicts the constructibility axiom V=L through increasingly sophisticated techniques involving ultrapower constructions. The second part focuses on the emergence of 0# from model-theoretic investigations. Following Ehrenfeucht and Mostowski's development of indiscernibility theory and its connections to Ramsey theory, Silver achieved a remarkable refinement by showing that the existence of 0# follows from the weaker assumption of a Ramsey cardinal. This report will systematically present the necessary background on the constructible hierarchy and measurable cardinals, culminating in a detailed proof of Silver's theorem that establishes the existence of 0# and its consequences.