The first week (Aug 5 - Aug 9):

王玮: 一阶算术的片段和模型.

The second week (Aug 12 - Aug 16):

Gabriel Goldberg: The Ultrapower Axiom.

Room: 3109. For Aug 16 (both lectures and sections), the room changed to HGX208 (光华楼西辅楼208).

Hours:

• Lecture 1: 9:15 - 10:30 (GMT+8)
• Lecture 2: 11:00 - 12:15 (GMT+8)
• Section: 15:00 - 17:00 (GMT+8)

一阶算术的片段和模型

本课程要求听众预先掌握以下基础知识: 一阶逻辑的完全性定理, 紧致性定理, 初等子模型的概念和判别初等子模型的 Tarski 准则, Lowenheim-Skolem 定理, Skolem 函数和 Skolem 闭包, 型 (type) 的概念, 哥德尔不完全性定理. 掌握上述定理意味着掌握它们的证明.

一阶算术的模型是一种类似 $(\mathbb{N},0,1,+,\times,<)$ 的数学结构. 通常我们期望这些数学结构满足一阶形式的皮亚诺算术公理 (简称 PA) 或其片段, 这时便可借助它们用数学方法研究数学的语言. 本课程计划先讨论一阶算术模型的内部结构、子结构, 再介绍构造新模型的方法, 然后介绍 PA 的一些片段以及如何用几种构造子模型或扩张的方法研究这些片段之间的关系. 如果时间允许, 我们将介绍 PA 的片段与组合数学的联系.

Program:

• Day 1: 语言、模型和公理
• Day 2: 子模型和扩张
• Day 3: Scott 集问题
• Day 4: PA 的片段
• Day 5: 一个组合定理的独立性

Reference

• Petr Hajek and Pavel Pudlak, Metamathematics of First-Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, 1993.
• Richard Kaye, Models of Peano Arithmetic, Oxford University Press, 1991.
• Roman Kossak and James Schmerl, The structure of models of Peano arithmetic. Oxford Logic Guides, 50, Oxford University Press, Oxford, 2006.

Lecturer:

王玮是中山大学哲学系教授。王玮的研究方向包括递归论、反推数学、算术模型论以及数理逻辑的其他分支。王玮教授曾在 Journal of Symbolic Logic、Advances in Mathematics、Transactions of the American Mathematical Society、Annals of Pure and Applied Logic 等期刊发表多篇论文。

The Ultrapower Axiom

This course will serve as an introduction to the theory of large cardinals and their inner models, the key concepts in the study of the consistency strength hierarchy beyond Zermelo-Fraenkel set theory. We begin with the theory of measurable, strongly compact, and supercompact cardinals. We then turn to the inner model theory of measurable cardinals and the problem of building inner models with larger cardinals. Finally, we discuss the Ultrapower Axiom and the prospects for an inner model with supercompact cardinals.

We assume the student is familiar with basic model theory, especially elementary embeddings and the ultraproduct construction. We also assume the following concepts from set theory: ordinals, cardinals, cofinalities, cardinal arithmetic, stationary sets, and the constructible universe.

Lecturer:

Gabriel Goldberg is an assistant professor at the department of mathematics, at UC Berkeley. He is working in set theory, interested in large cardinals, inner models, and infinite combinatorics. He has published some important papers in large cardinal and inner models in JEMS, JML and JSL in recent years. He has also published the book The Ultrapower Axiom. He has been awarded the Sacks Prize (2019), European Set Theory Plenary Speaker (2019), and Harvard Distinction in Teaching Award (2018).

• Zhaokuan Hao 郝兆宽
• Ruizhi Yang 杨睿之
• Ningyuan Yao 姚宁远
• Will Johnson

Contact: logic@fudan.edu.cn