# 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 B223**Organizers:**Uri Andrews and Hongyu Zhu

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 2023

The seminar will be run as a 1-credit seminar Math 975 in Fall 2023. If you are not enrolled but would like to audit it, please contact Uri Andrews and Hongyu Zhu.

While you are welcome (and encouraged) to present on a topic of your own choice, feel free to ask for help from faculties and/or other graduate students.

Presentation Schedule: https://docs.google.com/spreadsheets/d/15Qd4EzrrKpn1Ct5tur1P_FDc2czsdAVnUf_pfp65Lb4/edit?usp=sharing

Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)

Possible readings:

- (Elementary) Proof Theory: Chapters 4-7 of
*Aspects of Incompleteness*by Per Lindström. - An invitation to model-theoretic Galois theory.
*On arxiv here.* - Variations on the Feferman-Vaught Theorem
*On arxiv here.* - Any of several papers on "Turing Computable Embeddings"
- Computability/Model/Set Theory: Consult faculties/students for recommended texts on specific areas.

### September 11 - Organizational Meeting

We will meet to assign speakers to dates.

**September 18 - Taeyoung Em**

**Title:** Explicit construction of non-quasidetermined game on [math]\displaystyle{ \mathcal P(2^{\mathbb N}) }[/math] without using A.C. (Supplement)

**Abstract:** We will go over briefly some basic information about trees and infinite games. Then we prove the Gale-Stewart Theorem. The proof of the theorem motivates definition of quasistrategy. Then we will briefly introduce Borel determinacy. We will go over how the usage of A.C. makes convenient for us to make a non-quasidetermined or undertermined game. We will give an explicit construction of a non-quasidetermined game on [math]\displaystyle{ \mathcal P(2^{\mathbb N}) }[/math] without using A.C.

**September 25 - Karthik Ravishankar**

**Title:** Spectra of structures

**Abstract:** One way to measure the complexity of a structure is via its spectrum - the set of Turing degrees of its copies. In this talk, we'll look at the definition and first properties of the spectrum followed by some examples. In particular, we'll show that the non-computable degrees and the hyperimmune degrees form a spectrum while the DNC degrees do not.

**October 2 - Hongyu Zhu**

**Title:** Continuum Hypothesis: On Platonism and Pluralism (Slides and Recording; Passcode: .iXJs?1t)

**Abstract:** Despite its independence from ZFC, the continuum hypothesis continues to be of interest to logicians. In this talk, we will see arguments for settling the truth of CH in one way or another (or yet another). We will see how mathematical arguments (the inner model program) are intertwined with philosophical beliefs (mathematical Platonism and pluralism) about the set-theoretic universe(s).

**October 9 - Hannah Ashbach **

**Title:** An Introduction to Constructive Mathematics (Slides and Recording; Passcode: 5.$c5L+2)

**Abstract:** Have you ever written a mathematical proof and felt dissatisfied after writing QED? Perhaps you had proven the existence of a particularly complicated mathematical object, but you have no clue how that object may actually look or be constructed. Or perhaps you are ready to denounce the Axiom of Choice after reading about its far-reaching consequences. Constructive logic is a formal logic system that seeks to clear up these concerns for mathematicians, though not all mathematicians agree with the power it holds-- or takes away.

**October 16 - Rune Chen **

**Title:** An Introduction to Model-Theoretic Galois Theory

**Abstract:** Given an arbitrary first-order theory with elimination of imaginaries, the standard Galois theory can be translated into a model-theoretic version, which can be further generalized and thus be useful in other algebraic settings. In this talk, we will give a brief introduction to model-theoretic Galois theory and then look at a simple case of model-theoretic Galois correspondence. We will skip the detailed discussion of elimination of imaginaries by choosing our theory T to be a first-order theory coding finite sets and working in a large enough saturated model M of T.

**October 23 - John Spoerl **

**Title:** The Computational Content of Forcing

**Abstract:** Given information about a ground model, how much can you know about it’s forcing extensions? Given a forcing extension, how much can you know about its ground model? This talk will be a primer on forcing and a review of the main results of Hamkins, Miller and Williams’ 2020 paper: “Forcing as a Computational Process”.

**October 30 - Chiara Travesset **

**Title:** Mission-time LTL (MLTL) Formula Validation Via Regular Expressions

**Abstract:** Mission-time Linear Temporal Logic (mLTL) is an extension of propositional logic that includes temporal operators over finite intervals of time. A computation is a (finite) sequence consisting of a truth assignment for each propositional variable at each time step. This presentation will describe how we can use regular expressions to describe the structure of the computations that make a given mLTL formula true. We prove soundness and completeness, and also give an implemented algorithm (the WEST program) and analyze its complexity both theoretically and experimentally. We generate a test suite using control flow diagrams to robustly test the code. Finally, we present the REST theorem, which significantly simplifies certain sets of computations. This talk only requires basic familiarity with propositional logic and computer science.

**November 6 - Antonio Nakid Cordero **

**Title:** Martin's Conjecture in the Turing degrees and the Enumeration Degrees

**Abstract:** Martin's Conjecture is an attempt to formalize the empirical phenomenon that "naturally occurring" Turing degrees are well-ordered. In this talk, I'll go over how one can make that into a precise mathematical statement, a uniform version of the conjecture that has been fully resolved, and some of the complications of trying to propose a similar conjecture in the enumeration degrees. If time allows, I'll report on my ongoing project in this direction.

**November 13 - Alice Vidrine **

**Title:** Enumeration Weihrauch degrees and closed choice problems (Slides and Recording Passcode: C89k#5&0)

**Abstract:** In the Weihrauch degrees, we think of elements of Baire space as names for elements of mathematical objects. In the enumeration Weihrauch (eW) degrees we follow a similar idea, but our names may only carry positive information. It turns out that this coding by positive information separates several choice problems that are equivalent in the Weihrauch degrees, and in this talk we look at some of the more striking examples and their proofs. We close with some discussion of various directions for further research.

**November 20 - Logan Heath **

**Title:** A Proof Theorist's Guide to First Aid: CUT-Elimination and Other Results of Classical Proof Theory (Slides)

**Abstract:** When are correct proofs also "bad" proofs, how can we avoid them, and should we even try? In this talk we examine some early results of classical proof theory regarding the consistency of the sequent calculus for FOL and **PA**. In both cases our approach will be to avoid proofs which are technically correct, but also defective in some sense.

### November 27 - Thanksgiving Break

**December 4 - Mei Rose Connor **

**Title:** When ‘And’ is Too Much and ‘Or’ is Not Enough

**Abstract:** This work was motivated by the seven–valued predication system of the Jaina religion, which states that no statement exists as an absolute. This system, known as *Saptabhangivada* in Sanskrit, has 7 truth values that are the non-empty subsets of the set {asti, nāsti, avaktavyah}. These words translate to: *it is*, *it is not*, and *it is unassertible*, respectively. This talk will explore how these seven truth values relate to each other, the problems encountered when trying to formalise this system, and the author’s current resolution of said problems using lattice theory and introducing an eighth truth value.

**December 11 - NO MEETING**

Note: The originally scheduled talk has been moved to Tuesday, December 12 (at the usual logic seminar time&location). See https://people.math.wisc.edu/logic/seminar.html for more details.

## Previous Years

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