# 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:**Steffen Lempp 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

## Spring 2024

The seminar will be run as a 1-credit seminar Math 975 . In Spring 2024, the topic will be forcing constructions in computability theory. If you are not enrolled but would like to audit it, please contact Steffen Lempp and Hongyu Zhu.

Presentation Schedule: https://docs.google.com/spreadsheets/d/1JC6glG_soNLtaMQWaAuADlUu8dh2eJ0NL-MaUr7-nOk/edit?usp=sharing

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

### January 29 - Organizational Meeting

Steffen Lempp will give an overview and present some very basic forcing construction.

We will then assign speakers to dates and topics.

**February 5 - Taeyoung Em**

**Title:** Introduction to forcing

**Abstract:** We introduce new definitions and properties regarding forcing.

**February 12 - Hongyu Zhu**

**Title:** Slaman-Woodin Forcing and the Theory of Turing Degrees

**Abstract:** We will discuss how to use Slaman-Woodin forcing to interpret true second(first, resp.)-order arithmetic in the Turing degrees (Turing degrees below 0', resp.), thereby showing they have the same Turing degree.

**February 19 - John Spoerl**

**Title:** Forcing with Trees - Spector's and Sack's Minimal Degrees

**Abstract:** We'll take a look at Spector's forcing which uses perfect trees as conditions. Then we'll see where we might make some improvements which leads to Sack's sharpening of Spector's theorem: there is a minimal degree below 0'.

**February 26 - Karthik Ravishankar**

**Title:** The 3 element chain as an initial segment of the Turing Degrees

**Abstract:** In this talk, we'll look at the construction of a minimal degree with a strong minimal cover which shows that the three-element chain can be embedded as an initial segment of the Turing Degrees. The construction builds off ideas of Spector's minimal degree with stronger assumptions on the forcing conditions used. If time permits, we'll also talk about Copper's Jump Inversion building off Sack's construction.

**March 4 - Karthik Ravishankar**

**Title:** Bushy Tree forcing and constructing a minimal degree which is DNC

**Abstract:** We shall look at a forcing technique called Bushy Tree forcing using it to show that there is no uniform way to compute a DNC_2 from a DNC_3 function and that there is a DNC function that is weak in the sense that it does not compute a computably bounded DNC function. We present a few other results along these lines and sketch the construction of a minimal degree that is DNC relative to any given oracle using bushy tree forcing.

**March 11 - Josiah Jacobsen-Grocott**

**Title:** A uniformly e-pointed tree on Baire space without dead ends that is not of cototal degree

**Abstract:** A set is cototal if it is enumeration reducible to its complement. A tree is e-point if every path on the tree can enumerate the tree. McCathy proved that these notions are equivalent up to e-degree when considering e-pointed trees on cantor space. This fails when considering trees on Baire space. We give an example of a simple forcing construction that produces e-pointed trees on Baire space. We carefully analyze this forcing partial order to prove that generic e-pointed trees without dead ends are not of cototal degree.

**March 18 - Alice Vidrine**

**Title:** There is no non-computable bi-introreducible set

**Abstract:** A set is said to be bi-introreducible if it can be computed by any of its infinite subsets, or any infinite subset of its complement. This talk will detail a Matthias forcing construction used to prove a theorem by Seetapun which implies that the bi-introreducible sets are exactly the computable sets.

**April 1 - Hongyu Zhu**

**Title:** The Conservativeness of WKL_0 over RCA_0 for [math]\displaystyle{ \Pi_1^1 }[/math]-formulas

**Abstract:** We will see how to use forcing to construct models of WKL_0 from models of RCA_0 while preserving certain arithmetical truths, thereby showing that WKL_0 is [math]\displaystyle{ \Pi_1^1 }[/math]-conservative over RCA_0.

**April 8 - Cancelled**

**April 15 - Ang Li**

**Title:** Steel Forcing without Generalized Ramified Forcing Language

**Abstract:** In this talk, we will introduce Steel forcing, also known as "forcing with tagged trees", using a restricted infinitary language. We will prove the retagging lemma and show the example that Well-Foundededness and Unique Branch are not Borel separable. If time permits, we will talk about another example in descriptive set theory that uses steel forcing purely topologically.

**April 22 - Antonio Nakid Cordero**

**Title:** Kumabe-Slaman Forcing and Definability of the Turing Jump

**Abstract:** Slaman and Woodin's Analysis of the automorphisms of the Turing Degrees gave as a consequence the definability of the double jump. We will prove a Posner-Robinson type theorem for n-CEA operators by Kumabe-Slaman forcing that, together with the definability of the double jump, gives the definability of the single jump.

**April 29 - John Spoerl**

**Title:** Forcing in Set Theory

**Abstract:** Despite the theme of this seminar being about computability-theoretic forcing, the forcing method is most famously used in set theory to prove various consistency results. The most important use of forcing was its invention by Paul Cohen to show the independence of the continuum hypothesis and the axiom of choice from ZFC. I'll discuss the subtleties and development of forcing in set theory and sketch the method by which one can build and control new models of ZF(C).

## Previous Years

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