# Graduate Logic Seminar

The Graduate Logic Seminar is an informal space where graduate students and professors present topics related to logic which are not necessarily original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.

**When:**Mondays 3:30-4:30 PM**Where:**Van Vleck B139**Organizers:**Karthik Ravishankar and Antonio Nakid Cordero

The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.

Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu

## Fall 2022

### September 12 - Organizational Meeting

We will meet to assign speakers to dates.

**September 19 - Karthik Ravishankar**

**Title:** Lowness for Isomorphism (Slides)

**Abstract:** A Turing degree is said to be low for isomorphism if it can only compute an isomorphism between computable structures only when a computable isomorphism already exists. In this talk, we show that the measure of the class of low for isomorphism sets in Cantor space is 0 and that no Martin Lof random is low for isomorphism.

**September 26 - Antonio Nakid Cordero**

**Title:** When Models became Polish: an introduction to the Topological Vaught Conjecture (Slides)

**Abstract:** Vaught's Conjecture, originally asked by Vaught in 1961, is one of the most (in)famous open problems in mathematical logic. The conjecture is that a complete theory on a countable language must either have countably-many or continuum-many non-isomorphic models. In this talk, we will discuss some of the main ideas that surround this conjecture, with special emphasis on a topological generalization in terms of the continuous actions of Polish groups.

**October 10 - Yunting Zhang**

**Title:** Some History of Logic (Slides)

**Abstract:** The lives of great thinkers are sometimes overshadowed by their achievements-and there is perhaps no better illustration of this phenomenon than the life and work of Gödel. Take a look at Gödel's own timeline and see how wars and other mathematicians influenced him.

**October 17 - Alice Vidrine**

**Title:** Local operators, bilayer Turing reducibility, and enumeration Weihrauch degrees (Slides)

**Abstract:** Realizability toposes have a rich variety of subtoposes, corresponding to their local operators. These local operators are somewhat difficult to study in their usual form, which seems far removed from the usual objects of computability theoretic study. Recent work by Takayuki Kihara has given a characterization of the local operators on the effective topos in computability theoretic terms related to Weihrauch reduction, and which generalizes to several other realizability toposes of possible interest to computability theorists. This narrative-focused talk outlines what a realizability topos looks like, what local operators are, what Kihara's bilayer Turing reduction looks like, and how this leads to preliminary questions about a relative of the Weihrauch degrees based on enumeration reduction.

**October 24 - Hongyu Zhu**

**Title:** Investigating Natural Theories through the Consistency Operator (Slides)

**Abstract:** The phenomenon that "natural" theories tend to be linearly ordered in terms of consistency strength is a long-standing mystery. One approach to solving the problem is Martin's Conjecture, which roughly claims that the only natural functions on the Turing degrees are transfinite iterates of the Turing jump. In this talk we will focus on a similar approach, working inside the Lindenbaum algebra of elementary arithmetic instead of the Turing degrees. Here, the consistency operator takes the role of the jump. We will see that while some nice analogous claims can be established, there are also counterexamples that prevent us from strengthening the results in various ways.

**October 31 - Break for Halloween**

**November 7 - John Spoerl**

**Title:** Universal Algorithms

**Abstract:** Among the many ways we can flex Gödel’s Incompleteness theorems, there is one that feels especially strange: there is a partial computable function F, such that F gives no output in the standard model of arithmetic, but for any function G on natural numbers (computable or not) there is a non-standard model in which F behaves exactly as G. I’ll discuss some related arguments and some philosophical questions this may raise about our notions of finiteness and determinism.

**November 14 - Josiah Jacobsen-Grocott**

**Title:** The Paris-Harrington Theorem

**Abstract:** In this talk, I will present the proof of the Paris-Harrington theorem. The
Paris-Harrington theorem states that over PA the consistency of Peano arithmetic is equivalent to
a strengthening of the finite Ramsey theorem. This was the first example of a result from "ordinary
mathematics" that can not be proven by PA. The aim of this talk is to cover the main logical
steps in this proof and give some of the combinatorics.

**November 21 - Karthik Ravishankar**

**Title:** The computing power of Baire space vs Cantor Space

**Abstract:** Generic Muchnik reducibility extends the notion of Muchnik reducibility to the uncountable setting. In this talk we'll look at some recent work which comes up with another technique of constructing a structure of degree strictly between Cantor space and Baire Space. We'll see that there is a generic copy of Cantor space which computes no escaping function while every copy of Baire space computes a dominating function. We then construct a structure of intermediate degree which always computes an escaping function but no dominating function.

**November 28 - Logan Heath**

**Title**: A Mathematical Analysis of Theories of Generative Grammar: The Peters-Ritchie Theorem (Slides)

**Abstract**: This is a preview of a talk intended primarily for undergraduate students in linguistics who have just completed a first course in syntax. The goal of the talk is to spark interest in applications of mathematical techniques to the study of theories of syntax. We will focus on the Peters-Ritchie Theorem which shows that transformational grammars can generate languages which are c.e., but not computable.

**December 12 - Yuxiao Fu**

**Title:** Brief Introduction to Modal Logic (Slides)

**Abstract:** In this talk, I plan to give a mathematical introduction to modal logic and its applications, covering its syntax and semantics, Kripke frames and models, bisimulations, filtrations, frame definability, automatic first-order correspondence, canonical models, and conclude with the completeness of normal modal logics.

## Previous Years

The schedule of talks from past semesters can be found here.