Student Seminar on Computable Model Theory

Ren, Yvette 任晏黎 and Li, Yuxuan 李宇轩

Time: 11:00 am

Location: HGW2509

In 1949, Friedberg and Muchnik constructed two incomparable non-computable c.e. sets using priority arguments. Since then, priority methods have been developed and widely used to construct non computable objects. Various attempts have been made to give a general $0^{(n)}$-injury construction, including Harrington’s worker’s method, Lempp and Lerman’s trees of strategies, Ash–Knight’s $\eta$-system. In 2021, Montalbán gave a new metatheorem for Ash-Knight style priority constructions that is easier to use. The purpose of the seminar is to study the metatheorem, starting from basic computable structure seminar and priority arguments.

Program (tentative):

• Sept 19: Yvette Ren, Introduction to structures, presentations that code sets (CST: Within the Arithmetic chapter 1)
• Sept 26, Oct 10: Yuxuan Li, Relatively intrinsic computably enumerable relations (CST Within the Arithmetic chapter 2)
• Oct 17: Yvette Ren, Finite-injury constructions, priority arguments (CST Within the Arithmetic chapter 7)
• Oct 31, Nov 7: TBA, Ordinals and infinitary logic (CST Beyond the Arithmetic chapter 1,2)
• Nov 14: TBA, A new game metatheorem for Ash-Knight style priority constructions

Reference:

• Antonio Montalbán, Computable Structure Theory: Within the Arithmetic, 2021, Cambridge University Press.
• Antonio Montalbán, Computable Structure Theory: Beyond the Arithmetic, 2026, Cambridge University Press. Draft
• Antonio Montalbán, A New Game Metatheorem for Ash–Knight Style Priority Constructions, in Higher Recursion Theory and Set Theory, 2025. Draft