Set Theory Talks

Marko Malink (NYU) and Anubav Vasudevan (University of Chicago).
Title: The origins of conditional logic: Theophrastus on hypothetical syllogisms

Abstract: Łukasiewicz maintained that “the first system of propositional logic was invented about half a century after Aristotle: it was the logic of the Stoics”. In this talk, we argue that the first system of propositional logic was, in fact, developed by Aristotle’s pupil Theophrastus. Theophrastus sought to establish the priority of categorical over propositional logic by reducing various modes of propositional reasoning to categorical form. To this end, he interpreted the conditional “If φ then ψ” as a categorical proposition “A holds of all B”, in which B corresponds to the antecedent φ, and A to the consequent ψ. Under this interpretation, Aristotle’s law of subalternation (A holds of all B, therefore A holds of some B) corresponds to a version of Boethius’ Thesis (If φ then ψ, therefore not: If φ then not-ψ). Jonathan Barnes has argued that this consequence renders Theophrastus’ program of reducing propositional to categorical logic inconsistent. In this paper, we show that Barnes’s objection is inconclusive. We argue that the system developed by Theophrastus is both non-trivial and consistent, and that the propositional logic generated by Theophrastus’ system is exactly the connexive variant of the first-degree fragment of intensional linear logic. 

Speaker:     Saeed Salehi, University of Tabriz.

Date and Time:     Wednesday November 23, 2022, Zoom Talk SPECIAL TIME 9:30AM-11:00AM.

Title:     Self-Reference and Diagonalization: their difference and a short history.

Abstract: What is now called the Diagonal (or the Self-Reference) Lemma, is the statement that for every formula F(x), with the only free variable x, there exists a sentence σ such that σ is equivalent to the F of the Gödel code of σ, i.e., σ ≡ F(#σ); and this equivalence is provable in certain weak arithmetics. This lemma is credited to Gödel (1931), in the special case when F is the unprovability predicate, and to Carnap (1934) in the more general case.

In this talk, we will argue that Gödel-Carnap's original Diagonal Lemma is not the modern formulation and was more similar to, but not exactly identical with, the Strong Diagonal (or Direct Self-Reference) Lemma. This lemma, so-called recently, says that for every formula F(x), in a sufficiently expressive language, there exists a sentence σ such that σ is equal to the F of the Gödel code of σ, i.e., σ = F(#σ); and this equality is provable in sufficiently strong theories. We will attempt at tracking down the first appearance of the modern formulation of the Diagonal Lemma in the equivalent form, also in the strong direct form of equality.

Dima Sinapova, Rutgers University
Prikry sequences and square properties at ℵω

It is well known that if an inaccessible cardinal κ is singularized to countable cofinality while preserving cardinals, then □κω holds in the outer model. Moreover, this remains true even when relaxing the cardinal preservation assumption a bit. In this talk we focus on when Prikry forcing adds weaker forms of square in a more general setting. We prove abstract theorems about when Prikry forcing with interleaved collapses to bring down the singularized cardinal to ℵω will add a weak square sequence. This can be viewed as a partial positive result to a question of Woodin about whether the failure of SCH at ℵω implies weak square.

Marko Malink (NYU) and Anubav Vasudevan (University of Chicago).
Title: The origins of conditional logic: Theophrastus on hypothetical syllogisms

Abstract: Łukasiewicz maintained that “the first system of propositional logic was invented about half a century after Aristotle: it was the logic of the Stoics”. In this talk, we argue that the first system of propositional logic was, in fact, developed by Aristotle’s pupil Theophrastus. Theophrastus sought to establish the priority of categorical over propositional logic by reducing various modes of propositional reasoning to categorical form. To this end, he interpreted the conditional “If φ then ψ” as a categorical proposition “A holds of all B”, in which B corresponds to the antecedent φ, and A to the consequent ψ. Under this interpretation, Aristotle’s law of subalternation (A holds of all B, therefore A holds of some B) corresponds to a version of Boethius’ Thesis (If φ then ψ, therefore not: If φ then not-ψ). Jonathan Barnes has argued that this consequence renders Theophrastus’ program of reducing propositional to categorical logic inconsistent. In this paper, we show that Barnes’s objection is inconclusive. We argue that the system developed by Theophrastus is both non-trivial and consistent, and that the propositional logic generated by Theophrastus’ system is exactly the connexive variant of the first-degree fragment of intensional linear logic. 

Speaker:     Saeed Salehi, University of Tabriz.

Date and Time:     Wednesday November 23, 2022, TIME TBA.

Title:     Self-Reference and Diagonalization: their difference and a short history.

Abstract: What is now called the Diagonal (or the Self-Reference) Lemma, is the statement that for every formula F(x), with the only free variable x, there exists a sentence σ such that σ is equivalent to the F of the Gödel code of σ, i.e., σ ≡ F(#σ); and this equivalence is provable in certain weak arithmetics. This lemma is credited to Gödel (1931), in the special case when F is the unprovability predicate, and to Carnap (1934) in the more general case.

In this talk, we will argue that Gödel-Carnap's original Diagonal Lemma is not the modern formulation and was more similar to, but not exactly identical with, the Strong Diagonal (or Direct Self-Reference) Lemma. This lemma, so-called recently, says that for every formula F(x), in a sufficiently expressive language, there exists a sentence σ such that σ is equal to the F of the Gödel code of σ, i.e., σ = F(#σ); and this equality is provable in sufficiently strong theories. We will attempt at tracking down the first appearance of the modern formulation of the Diagonal Lemma in the equivalent form, also in the strong direct form of equality.

Hello everyone,

This week our weekly Nankai Logic Colloquium is going to be in the morning.

___________________________________________________________________________________________________________________________________________________

This is going to be an online event. Follow the link below to join the Zoom meeting. Please use your real name to join the meeting.

Title：                 The 9th Nankai Logic Colloquium -- Cesar E. Silva

Time：                9:00am, nov. 11, 2022 (Beijing Time)

Zoom Number：825 9529 2118

Passcode：         475473

Link：                  https://us02web.zoom.us/j/82595292118?pwd=ZHVGUG9FcUFlSWNqOEN0azlBMk1xdz09

_____________________________________________________________________

Best wishes,

Ming Xiao

Victoria Gitman (CUNY).
Title: Set theory without the powerset axiom

Abstract: Many natural and useful set-theoretic structures fail to satisfy the Powerset axiom. For example, the universe of sets can be decomposed into the H_alpha-hierarchy, indexed by cardinals alpha, where each H_alpha consists of all sets whose transitive closure has size less than alpha. If alpha is a regular cardinal, then H_alpha satisfies all axioms of ZFC except, maybe, the Powerset axiom (it will only satisfy Powerset if alpha is inaccessible). Class forcing extensions of models of ZFC will often fail to satisfy ZFC, but if the class forcing is nice enough, then it will preserve all the axioms of ZFC except, maybe, the Powerset axiom. Finally, a strong second-order set theory, extending Kelley-Morse by adding a choice principle for classes (Choice Scheme), is bi-interpretable with a strong first-order set theory without the Powerset axiom. Thus working in a strong enough second-order set theory can be reinterpreted as working in a strong first-order set theory in which the Powerset axiom fails. It turns out that simply taking the axioms of ZFC and removing the Powerset axiom does not yield a robust set theory. I will discuss robust (and strong) axiomatizations of set theory without Powerset and how much of the standard set theoretic machinery is still effective even in the strongest theories in the absence of Powerset. Because of the bi-interpretability of a strong set theory without Powerset with Kelley-Morse plus Choice Scheme, these results will have consequences for which set theoretic machinery continues to work in set theories with classes. Time permitting, I will also talk about some unexpectedly strange models of set theory without Powerset.

This will be another talk in the MOPA series on the history of the subject.

The work on generalized quantifiers in formal systems of arithmetic was initiated in 1980 by Macintyre, motivated by the search for natural extensions of first-order arithmetic that are immune to the Kirby-Paris-Harrington style independence results. Some open questions posed by Macintyre were solved in a definitive way in 1982 by Schmerl and Simpson and after that Schmerl wrote two more papers on for Peano Arithmetic in the languages with Ramsay stationary quantifiers. Some results of Macintyre were obtained independently by Carl Morgenstern. All these papers, while very well written, are quite technical and not easily accessible for readers who are not familiar with more advanced tools of the model theory of arithmetic. I will survey the results suppressing most technical details. I will also talk about an attempt to use logic with stationary quantifiers to classify ω1-like recursively saturated models of PA.

Asymmetric Cut and Choose Games

We consider the following two player game of infinite length: We are given a starting set X, and the players go by the names 'Cut' and 'Choose'. They take turns making moves, and in each step, Cut partitions a given set into two disjoint pieces, starting from the set X in their first move, and then Choose gets to pick one of the pieces, which is then partitioned into two pieces by Cut in their next move etc. In the end, Choose wins in case the intersection of all of their choices has at least two (distinct) elements.

We will investigate some of the properties of this game — in particular, we will discuss some classic results on when it is possible for one of the players to have a strategy for winning the game. We will then continue to discuss some variations of this game and their relevance to set theory — many central set theoretic notions, such as certain large cardinal properties, notions of distributivity, precipitousness and strategic closure were either known or turned out to be closely connected and often equivalent to the (non-)existence of winning strategies in certain cut and choose games.

This is joint work with Philipp Schlicht, Christopher Turner and Philip Welch (all University of Bristol).

Ali Enayat, University of Gothenburg
Tightness, solidity, and internal categoricity: Part II

Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

Corey Switzer, University of Vienna
The Special Tree Number

A tree of height ω1 with no cofinal branch is called special if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function f:T→ω so that if f(s)=f(t) then s and t are incomparable in the tree ordering. It is known that there is always a non-special tree of size continuum, but the existence of a smaller one is independent of ZFC. Motivated by this we introduce the special tree number, st, the least size of a tree of height ω1 which is neither non-special nor has a cofinal branch. Classical facts imply that st can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that st can be larger than a, g, and both the left hand side and bottom row of the Cichon diagram. Thus st is independent of many well known cardinal invariants. Central to this result is an in depth investigation of the types of reals added by the Baumgartner specialization poset which we will discuss as well.

Dave Marker, University of Illinois at Chicago
Automorphisms of differentially closed fields

Answering a question of Russell Miller, we show that there are differentially closed fields with no non-trivial automorphisms.

Victoria Gitman (CUNY).

Title: Set theory without the powerset axiom

Abstract: Many natural and useful set-theoretic structures fail to satisfy the Powerset axiom. For example, the universe of sets can be decomposed into the H_alpha-hierarchy, indexed by cardinals alpha, where each H_alpha consists of all sets whose transitive closure has size less than alpha. If alpha is a regular cardinal, then H_alpha satisfies all axioms of ZFC except, maybe, the Powerset axiom (it will only satisfy Powerset if alpha is inaccessible). Class forcing extensions of models of ZFC will often fail to satisfy ZFC, but if the class forcing is nice enough, then it will preserve all the axioms of ZFC except, maybe, the Powerset axiom. Finally, a strong second-order set theory, extending Kelley-Morse by adding a choice principle for classes (Choice Scheme), is bi-interpretable with a strong first-order set theory without the Powerset axiom. Thus working in a strong enough second-order set theory can be reinterpreted as working in a strong first-order set theory in which the Powerset axiom fails. It turns out that simply taking the axioms of ZFC and removing the Powerset axiom does not yield a robust set theory. I will discuss robust (and strong) axiomatizations of set theory without Powerset and how much of the standard set theoretic machinery is still effective even in the strongest theories in the absence of Powerset. Because of the bi-interpretability of a strong set theory without Powerset with Kelley-Morse plus Choice Scheme, these results will have consequences for which set theoretic machinery continues to work in set theories with classes. Time permitting, I will also talk about some unexpectedly strange models of set theory without Powerset.

This will be another talk in the MOPA series on the history of the subject.

The work on generalized quantifiers in formal systems of arithmetic was initiated in 1980 by Macintyre, motivated by the search for natural extensions of first-order arithmetic that are immune to the Kirby-Paris-Harrington style independence results. Some open questions posed by Macintyre were solved in a definitive way in 1982 by Schmerl and Simpson and after that Schmerl wrote two more papers on for Peano Arithmetic in the languages with Ramsay stationary quantifiers. Some results of Macintyre were obtained independently by Carl Morgenstern. All these papers, while very well written, are quite technical and not easily accessible for readers who are not familiar with more advanced tools of the model theory of arithmetic. I will survey the results suppressing most technical details. I will also talk about an attempt to use logic with stationary quantifiers to classify ω1-like recursively saturated models of PA.

Ali Enayat, University of Gothenburg
Tightness, solidity, and internal categoricity: Part II

Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

Corey Switzer, University of Vienna
The Special Tree Number

A tree of height ω1 with no cofinal branch is called special if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function f:T→ω so that if f(s)=f(t) then s and t are incomparable in the tree ordering. It is known that there is always a non-special tree of size continuum, but the existence of a smaller one is independent of ZFC. Motivated by this we introduce the special tree number, st, the least size of a tree of height ω1 which is neither non-special nor has a cofinal branch. Classical facts imply that st can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that st can be larger than a, g, and both the left hand side and bottom row of the Cichon diagram. Thus st is independent of many well known cardinal invariants. Central to this result is an in depth investigation of the types of reals added by the Baumgartner specialization poset which we will discuss as well.

Dave Marker, University of Illinois at Chicago
Automorphisms of differentially closed fields

Answering a question of Russell Miller, we show that there are differentially closed fields with no non-trivial automorphisms.

Victoria Gitman (CUNY).

Title: Set theory without the powerset axiom

Abstract: Many natural and useful set-theoretic structures fail to satisfy the Powerset axiom. For example, the universe of sets can be decomposed into the H_alpha-hierarchy, indexed by cardinals alpha, where each H_alpha consists of all sets whose transitive closure has size less than alpha. If alpha is a regular cardinal, then H_alpha satisfies all axioms of ZFC except, maybe, the Powerset axiom (it will only satisfy Powerset if alpha is inaccessible). Class forcing extensions of models of ZFC will often fail to satisfy ZFC, but if the class forcing is nice enough, then it will preserve all the axioms of ZFC except, maybe, the Powerset axiom. Finally, a strong second-order set theory, extending Kelley-Morse by adding a choice principle for classes (Choice Scheme), is bi-interpretable with a strong first-order set theory without the Powerset axiom. Thus working in a strong enough second-order set theory can be reinterpreted as working in a strong first-order set theory in which the Powerset axiom fails. It turns out that simply taking the axioms of ZFC and removing the Powerset axiom does not yield a robust set theory. I will discuss robust (and strong) axiomatizations of set theory without Powerset and how much of the standard set theoretic machinery is still effective even in the strongest theories in the absence of Powerset. Because of the bi-interpretability of a strong set theory without Powerset with Kelley-Morse plus Choice Scheme, these results will have consequences for which set theoretic machinery continues to work in set theories with classes. Time permitting, I will also talk about some unexpectedly strange models of set theory without Powerset.

This will be another talk in the MOPA series on the history of the subject.

The work on generalized quantifiers in formal systems of arithmetic was initiated in 1980 by Macintyre, motivated by the search for natural extensions of first-order arithmetic that are immune to the Kirby-Paris-Harrington style independence results. Some open questions posed by Macintyre were solved in a definitive way in 1982 by Schmerl and Simpson and after that Schmerl wrote two more papers on for Peano Arithmetic in the languages with Ramsay stationary quantifiers. Some results of Macintyre were obtained independently by Carl Morgenstern. All these papers, while very well written, are quite technical and not easily accessible for readers who are not familiar with more advanced tools of the model theory of arithmetic. I will survey the results suppressing most technical details. I will also talk about an attempt to use logic with stationary quantifiers to classify ω1-like recursively saturated models of PA.

Ali Enayat, University of Gothenburg
Tightness, solidity, and internal categoricity

Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

Ideal Independence, Filters and Maximal Sets of Reals

A family I⊆[ω]ω is called ideal independent if given any finite, distinct A,B0,...,Bn−1∈I, the set A∖⋃i<nBi is infinite. In other words, the ideal generated by I∖{A} does not contain A for any A∈I. The least size of a maximal (with respect to inclusion) ideal independent family is denoted smm and has recently been tied to several interesting questions in cardinal characteristics and Boolean algebra theory. In this talk we will sketch our new proof that this number is ZFC-provably greater than or equal to the ultrafilter number – the least size of a base for a non-principal ultrafilter on ω. The proof is entirely combinatorial and relies only on a knowledge of ultrafilters and their properties. Time permitting, we will also discuss some interesting new applications of ideal independent families to topology via a generalization of Mrowka spaces usually studied for almost disjoint families. This is joint work with Serhii Bardyla, Jonathan Cancino and Vera Fischer.

Ali Enayat, University of Gothenburg
Tightness, solidity, and internal categoricity: Part II

Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

Corey Switzer, University of Vienna
The Special Tree Number

A tree of height ω1 with no cofinal branch is called special if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function f:T→ω so that if f(s)=f(t) then s and t are incomparable in the tree ordering. It is known that there is always a non-special tree of size continuum, but the existence of a smaller one is independent of ZFC. Motivated by this we introduce the special tree number, st, the least size of a tree of height ω1 which is neither non-special nor has a cofinal branch. Classical facts imply that st can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that st can be larger than a, g, and both the left hand side and bottom row of the Cichon diagram. Thus st is independent of many well known cardinal invariants. Central to this result is an in depth investigation of the types of reals added by the Baumgartner specialization poset which we will discuss as well.

Dave Marker, University of Illinois at Chicago
Automorphisms of differentially closed fields

Answering a question of Russell Miller, we show that there are differentially closed fields with no non-trivial automorphisms.

Ali Enayat, University of Gothenburg
Tightness, solidity, and internal categoricity

Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

Ideal Independence, Filters and Maximal Sets of Reals

A family I⊆[ω]ω is called ideal independent if given any finite, distinct A,B0,...,Bn−1∈I, the set A∖⋃i<nBi is infinite. In other words, the ideal generated by I∖{A} does not contain A for any A∈I. The least size of a maximal (with respect to inclusion) ideal independent family is denoted smm and has recently been tied to several interesting questions in cardinal characteristics and Boolean algebra theory. In this talk we will sketch our new proof that this number is ZFC-provably greater than or equal to the ultrafilter number – the least size of a base for a non-principal ultrafilter on ω. The proof is entirely combinatorial and relies only on a knowledge of ultrafilters and their properties. Time permitting, we will also discuss some interesting new applications of ideal independent families to topology via a generalization of Mrowka spaces usually studied for almost disjoint families. This is joint work with Serhii Bardyla, Jonathan Cancino and Vera Fischer.

Ali Enayat, University of Gothenburg
Tightness, solidity, and internal categoricity: Part II

Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

Corey Switzer, University of Vienna
The Special Tree Number

A tree of height ω1 with no cofinal branch is called special if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function f:T→ω so that if f(s)=f(t) then s and t are incomparable in the tree ordering. It is known that there is always a non-special tree of size continuum, but the existence of a smaller one is independent of ZFC. Motivated by this we introduce the special tree number, st, the least size of a tree of height ω1 which is neither non-special nor has a cofinal branch. Classical facts imply that st can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that st can be larger than a, g, and both the left hand side and bottom row of the Cichon diagram. Thus st is independent of many well known cardinal invariants. Central to this result is an in depth investigation of the types of reals added by the Baumgartner specialization poset which we will discuss as well.

Dave Marker, University of Illinois at Chicago
Automorphisms of differentially closed fields

Answering a question of Russell Miller, we show that there are differentially closed fields with no non-trivial automorphisms.

Hello everyone,

This week our weekly Nankai Logic Colloquium is going to be in the afternoon.

Our speaker this week will be Jan Grebik from the University of Warwick. This talk is going to take place this Friday, Oct.28,  from 4 pm to 5 pm (UTC+8, Beijing time). 

Title: Complexity problem in measurable combinatorics

___________________________________________________________________________________________________________________________________________________

This is going to be an online event. Follow the link below to join the Zoom meeting. Please use your real name to join the meeting.

Title：                 The 8th Nankai Logic Colloquium -- Jan Grebik

Time：                16:00pm, Oct. 28, 2022 (Beijing Time)

Zoom Number：820 6148 1269

Passcode：         379819

Link：                  https://us02web.zoom.us/j/82061481269?pwd=TGkxT2Y3UUNCVUlKTUllZjhtMm1ZUT09

_____________________________________________________________________

Best wishes,

Ming Xiao

Kameryn Williams, Sam Houston University
Tightness in second-order arithmetic

Say that a theory T is tight if any two distinct extensions of T cannot be bi-interpretable. Vaguely speaking, tightness expresses a sort of maximality to the expressiveness of T. Visser showed that PA is tight and building on this work, Enayat showed that Z2, second-order arithmetic with full second-order comprehension, is also tight. In this talk I will address the question of whether full logical strength of these theories of arithmetic are necessary to have tightness, focusing on subsystems of Z2. The answer to this question is positive. If you restrict the comprehension axiom of Z2 to only arithmetical formulae, or if you restrict it to Σ1k formulae, the resulting theory is not tight. As a specific instance, we show that if M is either the minimum omega-model of ACA0 or the minimum beta-model of Π1k-CA for some k≥1, then M is bi-interpretable with a carefully chosen extension M[c] by Cohen-forcing.

This talk is about joint work with Alfredo Roque Freire.

Computational Logic Seminar
Fall Semester 2022
Tuesday, October 18
Time 2:00 - 4:00 PM
Room 3310-B
Speaker: Tudor Protopopescu, CUNY
Title: Intuitionistic Epistemic Logic

Abstract:
We outline an intuitionistic view of knowledge which maintains the original Brouwer-Heyting-Kolmogorov semantics for intuitionism and is consistent with the well-known approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view co-reflection A -> KA is valid and the factivity of knowledge holds in the form KA -> ~~A `known propositions cannot be false'. We show that the traditional form of factivity KA -> A is a distinctly classical principle which, like tertium non datur A v ~A, does not hold intuitionistically, but, along with the whole of classical epistemic logic, is intuitionistically valid in its double negation form ~~(KA -> A). Within the intuitionistic epistemic framework, the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed to it.

Speaker:     David Ellerman, University of Ljubljana.

Date and Time:     Wednesday October 19, 2022, 7:00 - 8:30 PM.

Title:     To Interpret Quantum Mechanics:``Follow the Math'': The math of QM as the linearization of the math of partitions.

Abstract: Set partitions are dual to subsets, so there is a logic of partitions dual to the Boolean logic of subsets. Partitions are the mathematical tool to describe definiteness and indefiniteness, distinctions and distinctions, as well as distinguishability and indistinguishability. There is a semi-algorithmic process or ``Yoga'' of linearization to transform the concepts of partition math into the corresponding vector space concepts. Then it is seen that those vector space concepts, particularly in Hilbert spaces, are the mathematical framework of quantum mechanics. (QM). This shows that those concepts, e.g., distinguishability versus indistinguishability, are the central organizing concepts in QM to describe an underlying reality of objective indefiniteness--as opposed to the classical physics and common sense view of reality as ``definite all the way down'' This approach thus supports what Abner Shimony called the ``Literal Interpretation'' of QM which interprets the formalism literally as describing objective indefiniteness and objective probabilities--as well as being complete in contrast to the other realistic interpretations such as the Bohmian, spontaneous localization, and many world interpretations which embody other variables, other equations, or other worldly ideas.

The underlying paper is forthcoming in the Foundations of Physics, and the preprint is in the ArXiv here.

Forcing 'NSω1 is ω1-dense' from Large Cardinals - A Journey guided by the Stars

An ideal I on ω1 is ω1-dense if (P(ω1)/I)+ has a dense subset of size ω1. We prove, assuming large cardinals, that there is a semiproper forcing P so thatVP⊨‘NSω1 is ω1-dense'.This answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of Pmax in Woodin's axiom (∗) by Qmax results in an axiom Qmax−(∗) which implies ‘NSω1 is ω1-dense'.
We proceed in three steps: First we define and motivate a new forcing axiom QM and then modify the Asperó-Schindler proof of ‘MM++⇒(∗)' to show ‘QM⇒Qmax−(∗)'. Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing QM. This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

Over half a century ago Hyman Bass proved that all flat left modules are projective precisely when the underlying ring satisfies the descending chain condition on right principal ideals. He called such rings left perfect. Gena Puninski noticed that this can be given a model theoretic proof. Every infinite descending chain of principal right ideals gives rise to a descending chain of (pp) formulas which, in turn, gives rise to a direct limit of finitely generated projective modules that is not projective. Such a module is flat and not projective, and called a Bass module.

I demonstrate how this construction is elementary model theory and at the same time generalizes to other classes of (pp) formulas and modules, which, among other things, yields a new proof of the late Daniel Simson’s result that all left modules are Mittag-Leffler iff the ring is left pure-semisimple (which, to model theorists, means that all left modules are totally transcendental).

I will emphasize the model theoretic ideas and explain the connection with the algebraic concepts. This is part of ongoing work with Anand Pillay.

The computability of Artin-Rees and Krull Intersection

We will explore the computational content of two related algebraic theorems, namely the Artin-Rees Lemma and Krull Intersection Theorem. In particular we will show that, while the strengths of these theorems coincide for individual rings, they become distinct in the uniform context.

Kameryn Williams, Sam Houston University
Tightness in second-order arithmetic

Say that a theory T is tight if any two distinct extensions of T cannot be bi-interpretable. Vaguely speaking, tightness expresses a sort of maximality to the expressiveness of T. Visser showed that PA is tight and building on this work, Enayat showed that Z2, second-order arithmetic with full second-order comprehension, is also tight. In this talk I will address the question of whether full logical strength of these theories of arithmetic are necessary to have tightness, focusing on subsystems of Z2. The answer to this question is positive. If you restrict the comprehension axiom of Z2 to only arithmetical formulae, or if you restrict it to Σ1k formulae, the resulting theory is not tight. As a specific instance, we show that if M is either the minimum omega-model of ACA0 or the minimum beta-model of Π1k-CA for some k≥1, then M is bi-interpretable with a carefully chosen extension M[c] by Cohen-forcing.

This talk is about joint work with Alfredo Roque Freire.

Over half a century ago Hyman Bass proved that all flat left modules are projective precisely when the underlying ring satisfies the descending chain condition on right principal ideals. He called such rings left perfect. Gena Puninski noticed that this can be given a model theoretic proof. Every infinite descending chain of principal right ideals gives rise to a descending chain of (pp) formulas which, in turn, gives rise to a direct limit of finitely generated projective modules that is not projective. Such a module is flat and not projective, and called a Bass module.

I demonstrate how this construction is elementary model theory and at the same time generalizes to other classes of (pp) formulas and modules, which, among other things, yields a new proof of the late Daniel Simson’s result that all left modules are Mittag-Leffler iff the ring is left pure-semisimple (which, to model theorists, means that all left modules are totally transcendental).

I will emphasize the model theoretic ideas and explain the connection with the algebraic concepts. This is part of ongoing work with Anand Pillay.

Athar Abdul-Quader, Purchase College
Pathologically defined subsets of models of CT−

It is well known that every countable recursively saturated model of PA has a full compositional truth predicate; that is, such a model is expandable to the theory CT−. It is also well known that such a truth predicate need not be inductive, or indeed, need not satisfy even Δ0 induction. Recently, Enayat and Pakhomov showed that Δ0 induction for the truth predicate is equivalent to the principle of disjunctive correctness: the assertion that for any sequence of sentences ⟨ϕi:i<c⟩, the disjunction ⋁i<cϕi is evaluated as true if and only if there is i<c such that ϕi is evaluated as true. In the absence of Δ0 induction, various pathologies can occur, including models of CT− for which all nonstandard length disjunctions are evaluated as true. In this talk, we classify the sets X for which there is a model of CT− in which X is exactly the set of those c such that the disjunctions of length c of 0 = 1 is evaluated as false. In particular, we see that X can be ω if and only if ω is a strong cut, and therefore the 'disjunctively trivial' models mentioned before are in fact arithmetically saturated. This is joint work (in progress) with Mateusz Łełyk.

Definability of Laver-generic large cardinals and largeness of generic large cardinals with chain conditions

For a class P of posets, a cardinal κ is said to be generically supercompact by P (or P-gen. supercompact for short) if, for any λ≥κ, there are P∈P such that, for all (V,P)-generic G there are j, M⊆V[G] with j:V≺→κM, j(κ)>λ, and j''λ∈M.

A cardinal κ is Laver-generically supercompact for P (or P-Laver-gen. supercompact for short) if, for any λ≥κ, P∈P and (V,P)-generic G, there are P-name ˙Q with ⊩P′′˙Q∈P′′ such that, for all (V,P∗˙Q)-generic H⊇G, there are j, M⊆V[H] such that j:V≺→κM, j(κ)>λ, and P∗˙Q, H, j''λ∈M.