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Mathematical Logic at Fudan

2021 Fudan Logic Summer School

Date: Jun 21 - 25, Jul 26 - 30, and Aug 2 - 6

Location: Tencent Meeting / Zoom Meeting (Meeting ID To be send via email according to the registration)


The first week (Jun 21 - Jun 25):

宋诗畅 Shichang Song: Continuous First Order Logic (in Chinese)

  • Lecture1: 9:00 - 10:15 (GMT+8)
  • Lecture2: 10:45 - 12:00 (GMT+8)
  • Section: 14:00 - 16:00 (GMT+8)

The second week (Jul 26 - Jul 30):

Rehana Patel: Probabilistic Constructions in Model Theory

  • Lecture 1: 14:30 - 15:45 (GMT+8)
  • Lecture 2: 16:15 - 17:30 (GMT+8)
  • Section: (Next day) 9:30 - 11:30 (GMT+8)

The third week (Aug 2 - Aug 6):

Antonio Montalban: Scott complexity of countable structures

  • Lecture 1: 8:00 - 9:15 (GMT+8)
  • Lecture 2: 9:45 - 11:00 (GMT+8)
  • Section: 14:00 - 16:00 (GMT+8)

Week 1

Continuous First Order Logic

本课程讲述的连续一阶逻辑(continuous first order logic),有时候也叫做度量结构的模型论(model theory for metric structures),连续模型论,或者就叫连续逻辑。它是由Ben Yaacov, Berenstein, Henson, Usvyatsov在2008年前后发展起来的一种多值逻辑。连续一阶逻辑跟经典一阶逻辑最大的不同是,连续一阶逻辑的真值表是整个[0,1]区间。连续模型论作为模型论的一种推广,保持了很多模型论的特性,比如,连续模型论满足,紧致性定理,Lowenheim-Skolem定理,可以定义型空间,讨论量词消解,范畴性和稳定性。本课程将从连续一阶逻辑的语法和语义出发,详细地讲述连续一阶逻辑的基础知识。最后,作为例子,介绍连续一阶逻辑在概率论中的应用。本课程只需要基础的一阶逻辑知识。

The lectures will be in Chinese.


  • I. Ben Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov, Model theory for metric structures, in: Model Theory with Applications to Algebra and Analysis, Volume 2, London Math. Society Lecture Note Series, 350, Cambridge University Press, 2008, 315-427.
  • I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Amer. Math. Soc. 362 (2010), 5213-5259.


  • Day 1: Metric structures, signatures, formulas, semantics. Notes Video
  • Day 2: Ultraproducts, Compactness Theorem, connectives. Notes Video
  • Day 3: Lowenheim-Skolem Theorem, types, definability. Notes Video
  • Day 4: Omitting types theorem, separably categoricity, quantifier elimination, stability. Notes Video
  • Day 5: Application to probability theory; probability algebras, random variable structures. Notes Video


宋诗畅,北京交通大学,数学系,副教授。博士毕业于伊利诺伊大学香槟分校。曾在中国科学院数学研究所从事博士后研究。学术研究领域是模型论及其在概率论,群论和组合数学的应用。在Fund. Math., MLQ Math. Log. Q., J. Korean Math. Soc.杂志发表论文多篇。

Week 2

Probabilistic Constructions in Model Theory

Probabilistic constructions of countable objects play an important role in combinatorics and logic. Perhaps the most well known such construction is the Erdos-Rényi process, in which edges on pairs of distinct natural numbers are determined by independent tosses of a fair coin, producing the Rado graph up to isomorphism with probability 1. The Erdos-Rényi process has the symmetry property of “exchangeability", that is, it does not depend on the order in which the pairs of distinct natural numbers are considered. These lectures will aim to give students a taste of probabilistic constructions that arise in model theory, with a focus on ones that are exchangeable. Topics will include first order zero-one laws, invariant measures via sampling from Borel structures, and sparse random graphs; these may be modified given time constraints and student interest.


  • N. Ackerman, C. Freer, and R. Patel, Invariant measures concentrated on countable structures, Forum Math. Sigma 4 (2016), no. e17, 59 pp.
  • J. T. Baldwin and S. Shelah, Randomness and semigenericity, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1359–1356.
  • K.J. Compton, “Laws in logic and combinatorics”, in: I. Rival (ed.), Algorithms and order, NATO ASI Series (Series C: Mathematical and Physical Sciences), vol. 255, Springer, Dordrecht, 1989, 353–383.
  • R. Fagin, Probabilities on finite models, J. Symb. Log. 41 (1976), no. 1, 50–58.
  • H. Gaifman, Concerning measures in first order calculi, Israel J. Math. 2 (1964), 1–18.
  • Yu.V. Glebskii, D.I. Kogan, M.I. Liogon’kii, and V.A. Talonov, Range and degree of realizability of formulas in the restricted predicate calculus, Cybernet. Systems Anal. 5 (1969), no. 2, 142–154.
  • W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
  • P.G. Kolaitis, H.J. Prömel, and B.L. Rothschild, Kl+1-free graphs: asymptotic structure and a 0-1 law, Trans. Amer. Math. Soc. 303 (1987), no. 2, 637–671.
  • F. Petrov and A. Vershik, Uncountable graphs and invariant measures on the set of universal countable graphs, Random Structures Algorithms 37 (2010), no. 3, 389–406.
  • J. Spencer, The strange logic of random graphs, Algorithms and Combinatorics, vol. 22, Springer, Berlin, 2001.


  • Day 1: Model-theoretic preliminaries, Fraissé theory, and the Erdos-Rényi process.
  • Day 2: First order zero-one laws for combinatorial classes. Notes
  • Day 3: Exchangeable constructions of countable structures. Notes
  • Day 4: Exchangeable constructions of first-order theories. Notes
  • Day 5: Random constructions of sparse Hrushovski structures. Notes


Rehana Patel is a model theorist. Her main research interests are in applications of model theory to the study of combinatorics and random structures. She has held faculty positions at Harvard University, Olin College of Engineering, and other institutions in the United States, and is a visiting lecturer at the African Institute for Mathematical Sciences, Senegal.

Week 3

Scott complexity of countable structures

The lectures will be based on the following books with concentration on the Chapter II of the second book.

Slides Slides with notes Video


  • Day 1: Overview of the notions and the main results.
  • Day 2: The Infinitary language.
  • Day 3: (break)
  • Day 4: The back-and-forth relations.
  • Day 5: Computable categoricity.


Antonio Montalban is Professor at the department of mathematics, at the University of California, Berkeley.