Difference between revisions of "Graduate Logic Seminar"

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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.
 
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:''' Tuesdays 4-5 PM
+
* '''When:''' Mondays 3:30-4:30 PM
* '''Where:''' Van Vleck 901
+
* '''Where:''' Van Vleck B139
* '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh]
+
* '''Organizers:''' Karthik Ravishankar and [https://sites.google.com/wisc.edu/antonio 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.
 
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.
Line 9: Line 9:
 
Sign up for the graduate logic seminar mailing list:  join-grad-logic-sem@lists.wisc.edu
 
Sign up for the graduate logic seminar mailing list:  join-grad-logic-sem@lists.wisc.edu
  
== Spring 2022 ==
+
== Fall 2022 ==
  
The graduate logic seminar this semester will be run as MATH 975. Please enroll if you wish to participate.
+
=== September 12 - Organizational Meeting ===
 
 
We plan to cover the first 9 parts of [https://blog.nus.edu.sg/matwong/teach/modelarith/ Tin Lok Wong's notes], as well as a few other relevant topics which are not covered in the notes:
 
* Properness of the induction/bounding hierarchy (chapter 10 of Models of Peano Arithmetic by Kaye is a good source)
 
* Tennenbaum's theorem (this is a quick consequence of the main theorem of part 4, so it should be combined with part 4 or part 5)
 
* Other facts found in chapter 1 of [http://homepages.math.uic.edu/~marker/marker-thesis.pdf David Marker's thesis].
 
 
 
=== January 25 - organizational meeting ===
 
  
 
We will meet to assign speakers to dates.
 
We will meet to assign speakers to dates.
  
=== February 1 - Steffen Lempp ===
+
=== '''September 19 - Karthik Ravishankar''' ===
 
+
'''Title:''' Lowness for Isomorphism ([https://wiki.math.wisc.edu/images/Karthik_talk.pdf Slides])
I will give an overview of the topics we will cover:
 
 
 
1. the base theory PA^- and the induction and bounding axioms for Sigma_n-formulas, and how they relate to each other,
 
 
 
2. the equivalence of Sigma_n-induction with a version of Sigma_n-separation (proved by H. Friedman),
 
 
 
3. the Grzegorczyk hierarchy of fast-growing functions,
 
 
 
4. end extensions and cofinal extensions,
 
 
 
5. recursive saturation and resplendency,
 
 
 
6. standard systems and coded types,
 
 
 
7. the McDowell-Specker Theorem that every model of PA has a proper elementary end extension, and
 
 
 
8. Gaifman's theorem that every model of PA has a minimal elementary end extension.
 
 
 
I will sketch the basic definitions and state the main theorems, in a form that one can appreciate without too much
 
background.
 
 
 
=== February 8 - Karthik Ravishankar ===
 
 
 
Title: Collection axioms
 
 
 
We will discuss parts 1 and 2 of Wong's notes.
 
 
 
=== February 15 - Karthik Ravishankar, Yunting Zhang ===
 
 
 
Title: Collection axioms/The Weak König Lemma
 
 
 
Karthik will finish part 2 of Wong's notes. Then Yunting will start on part 3 of Wong's notes.
 
  
=== February 22 - Yunting Zhang ===
+
'''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.
  
Title: The Weak König Lemma
+
=== '''September 26 - Antonio Nakid Cordero'''  ===
 +
'''Title:''' When Models became Polish: an introduction to the Topological Vaught Conjecture ([[Media:GradLogSem - Topological Vaught Conjecture.pdf|Slides]])
  
We will finish part 3 of Wong's notes.
+
'''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.
  
=== March 22 - Ang Li ===
+
=== '''October 10 - Yunting Zhang''' ===
 +
'''Title:''' Some History of Logic ([[Media:Godel.pdf|Slides]])
  
Title: The Arithmetized Completeness Theorem
+
'''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.
  
We will discuss part 4 of Wong's notes.
+
=== '''October 17 - Alice Vidrine''' ===
 +
'''Title:''' Local operators, bilayer Turing reducibility, and enumeration Weihrauch degrees ([[Media:LTeW-talk.pdf|Slides]])
  
=== March 29 - Ang Li ===
+
'''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.
  
Title: The Arithmetized Completeness Theorem
+
=== '''October 24 - Hongyu Zhu''' ===
 +
'''Title:''' Investigating Natural Theories through the Consistency Operator ([[Media:ConsistencyOperator.pdf|Slides]])
  
We will finish part 4 of Wong's notes.
+
'''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.  
  
=== April 5 - Antonio Nákid Cordero ===
+
=== '''October 31 - Break for Halloween''' ===
  
Title: Semiregular cuts
+
=== '''November 7 - John Spoerl''' ===
 +
'''Title:''' Universal Algorithms
  
We will start on part 5 of Wong's notes.
+
'''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.
  
=== April 12 - Antonio Nákid Cordero/Alice Vidrine ===
+
=== '''November 14 - Josiah Jacobsen-Grocott''' ===
 +
'''Title:''' The Paris-Harrington Theorem
  
Title: Semiregular cuts/End and cofinal extensions
+
'''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.
  
We will finish part 5 of Wong's notes and then start on part 6.
+
=== '''November 21 - Karthik Ravishankar''' ===
 +
'''Title:''' The computing power of Baire space vs Cantor Space
  
=== April 19 - Alice Vidrine ===
+
'''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.
  
Title: End and cofinal extensions
+
=== '''November 28 - Logan Heath''' ===
  
We will finish part 6 of Wong's notes.
+
=== '''December 5 - Logan Heath''' ===
  
=== May 3 - No seminar today ===
+
=== '''December 12 - Yuxiao Fu''' ===
  
 
== Previous Years ==
 
== Previous Years ==
  
 
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].
 
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].

Latest revision as of 15:57, 21 November 2022

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

December 5 - Logan Heath

December 12 - Yuxiao Fu

Previous Years

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