## (KGRC) two talks in the Set Theory Seminar on Tuesday, December 6

## (KGRC) talks Tuesday, November 29 and Wednesday, November 30

## Wednesday seminar

## Cross-Alps Logic Seminar (speaker: Francesco Parente)

**Francesco Parente**(University of Turin)

will give a talk on

*Good ultrafilters and universality properties of forcing*Please refer to the usual webpage of our LogicGroup for more details and the abstract of the talk.

The seminar will be held remotely through Webex. Please write to vincenzo.dimonte [at] uniud [dot] it for the link to the event.

The Cross-Alps Logic Seminar is co-organized by the logic groups of Genoa, Lausanne, Turin and Udine as part of our collaboration in the project PRIN 2017 'Mathematical logic: models, sets, computability'.

## Nankai Logic Colloquium

Hello everyone,

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

Title: Extremal models in affine logic Abstract: Affine logic is a fragment of continuous logic, introduced by Bagheri, where one allows only affine functions R^n -> R as connectives instead of arbitrary continuous functions. This decreases the expressive power of the logic and provides additional structure on the type spaces: namely, the structure of compact, convex sets. An important role in convex analysis is played by the extreme points of these sets and, unsurprisingly, extremal models, in which only extreme types are realized, are crucial for developing affine model theory. In a joint work with Itaï Ben Yaacov and Tomás Ibarlucía, we develop the basic theory of extremal models. Some highlights include a general integral decomposition theorem (generalizing the ergodic decomposition theorem from ergodic theory) and affine aleph_0-categoricity: theories admitting a unique separable, extremal model. In the talk, I will give a gentle introduction to affine logic and will explain some of our main results.

___________________________________________________________________________________________________________________________________________________

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 11th Nankai Logic Colloquium -- Todor Tsankov

Time： 16:00pm, Dec. 2, 2022 (Beijing Time)

Zoom Number：817 7281 1616

Passcode： 468722

Link： https://us02web.zoom.us/j/81772811616?pwd=ZGU2UHlSbEYzL3lQOUk0YzFPeTRLUT09

_____________________________________________________________________

Best wishes,

Ming Xiao

## This Week in Logic at CUNY

Date: Monday, November 28, 4.15-6.15 (NY time), GC 7314

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

William McCarthy (Columbia).

Title: Modal pluralism and higher-order logic

Abstract: Modal pluralism is the view that there are a variety of candidate interpretations of the predicate ‘could have been the case that’ which give intuitively different answers to paradigmatic metaphysical questions (‘intuitively’ because the phrase means subtly different things on the different interpretations). It is the modal analog of set-theoretic pluralism, according to which there are a variety of candidate interpretations of ‘is a member of’. Of course, if there were a broadest kind of counterfactual possibility, then one could define every other kind as a restriction on it, as in the set-theoretic case. It would then be privileged in the way that a broadest kind of set would be, if there were one. Recently, several authors have purported to prove from higher-order logical principles that there is a broadest kind of possibility. In this talk we critically assess these arguments. We argue that they rest on an assumption which any modal pluralist should reject: namely, monism about higher-order logic. The reasons to be a modal pluralist are also reasons to be a pluralist about higher-order quantification. But from the pluralist perspective on higher-order logic, the claim that there is a broadest kind of possibility is like the Continuum Hypothesis, according to the set-theoretic pluralist. It is true on some interpretations of the relevant terminology, and false on others. Consequently, the significance of the ‘proof’ that there is a broadest kind of possibility is deflated. Time permitting, we will conclude with some upshots of higher-order pluralism for the methodology of metaphysics.

Note: This is joint work with Justin Clarke-Doane.

- - - - Tuesday, Nov 29, 2022 - - - -

- - - - Wednesday, Nov 30, 2022 - - - -

- - - - Thursday, Dec 1, 2022 - - - -

- - - - Friday, Dec 2, 2022 - - - -

CUNY Graduate Center, Friday, December 2, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

CUNY Graduate Center

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

Friday December 2, 2:00pm-3:30pm, Room 6417

- - - - Monday, Dec 5, 2022 - - - -

Determinacy in the Chang model from a hod pair

Logic and Metaphysics Workshop

Date: Monday, November 28, 4.15-6.15 (NY time), GC 7314

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: Reification as identity?

Abstract: Abstract objects like properties and propositions, I believe, are the result of reification, which can intuitively be characterized as the metaphysical counterpart of nominalization (as in the shift, e.g., from ‘is a horse’ to ‘the property of being a horse’; cf. Schiffer, Moltmann), and occurs paradigmatically in the well-known bridge laws for instantiation, truth, etc. (e.g., something instantiates the property of being a horse iff it is a horse). So far, I have been working on an account of reification in terms of the technical notions of encoding & decoding, as some regulars at the L+M workshop may recall. In my upcoming talk, I wish to embed reification more clearly in higher-order metaphysics and explore an alternative idea: Can reification be construed as identification across metaphysical categories? E.g., can the object that is the property of being a horse be identified, in some sense, with Frege’s concept horse, which is a non-objectual item because ‘is a horse’ is not a singular term? In my presentation I will argue for an affirmative answer. For this, I will sketch an ultra-generalized logic of equivalence, which has as its special cases (i) the well-known logics of first-order identity and equivalence, (ii) recent logics of generalized identities (à la Rayo, Linnebo, Dorr, Fine, Correia, Skiles, …) which connect higher-order items of the same type, and (iii) the logic of my proposed cross-level equivalences which connect items of different types. In a second step, I will re-construe reification as the cross-level equivalence that holds between higher-order items and abstract objects of the appropriate sort and argue that this account of reification as identity has certain advantages.

- - - - Tuesday, Dec 6, 2022 - - - -

- - - - Wednesday, Dec 7, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York

Speaker: ** Robert Pare, Dalhousie University.**

Date and Time: ** Wednesday December 7, 2022, 7:00 - 8:30 PM.**

Title:** The horizontal/vertical synergy of double categories.**

Abstract: A double category is a category with two types of arrows, horizontal and vertical, related by double cells. Think of sets with functions and relations as arrows and implications as double cells. The theory is 2-dimensional just like for 2-categories. In fact 2-categories were originally defined as double categories in which all vertical arrows were identities. Most of the theory of 2-categories extends to double categories resulting in a deeper understanding. This is one aspect of double categories: they’re “new and improved” 2-categories.

From a purely formal point of view, a double category is a category object in CAT. Once a familiarity with double categories has developed, it is amusing and instructive to see how the various constructs of formal category theory play out in this setting.

But these two aspects of double categories, fancy 2-categories or internal categories, are only part of the picture. Perhaps the most important thing is the interplay between the horizontal and the vertical.

I will start with some examples of double categories to give a feeling for the objects I will be discussing, and then look at several concepts indicative of the rich interplay between the horizontal and the vertical.

- - - - Thursday, Dec 8, 2022 - - - -

- - - - Friday, Dec 9, 2022 - - - -

CUNY Graduate Center, Friday, December 9, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

CUNY Graduate Center

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

Friday December 9, 2:00pm-3:30pm, Room 6417

In this talk, I will give an overview of the picture of the Borel Wadge degrees. Our system of descriptions allows us to describe their Delta-classes, as well as specify which degrees have the separation or reduction properties. Part of our analysis is based on playing games along our descriptions, and so I will explain how these games are played and what they can tell us.

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Core model seminar on Tuesday

## Nankai Logic Colloquium

## This Week in Logic at CUNY

Logic and Metaphysics Workshop

Date: Monday, November 21, 4.15-6.15 (NY time), GC 7314

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

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.

Simon Thomas, Rutgers

Invariant Random Subgroups and Characters

- - - - Tuesday, Nov 22, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 22, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

**Joel David Hamkins**, University of Notre Dame**Pointwise definable and Leibnizian extensions of models of arithmetic and set theory**

I shall introduce a flexible new method showing that every countable model of PA admits a pointwise definable end-extension, one in which every individual is definable without parameters. And similarly for models of set theory, in which one may also achieve the Barwise extension result—every countable model of ZF admits a pointwise definable end-extension to a model of ZFC+V=L, or indeed any theory arising in a suitable inner model. A generalization of the method shows that every model of arithmetic of size at most continuum admits a Leibnizian extension, and similarly in set theory.

**Computational Logic Seminar**

**Fall Semester 2022**

**Tuesday, November 22**

Time 2:00 - 4:00 PM

Time 2:00 - 4:00 PM

**Room 3310-B,**

**The talk will be delivered online for a live audience,**

**for a zoom link contact**m

**Speaker**:

*Neil De Boer, The Ohio State University*

**Title:**

*Justification Logic and Type Theory as Formalizations of Intuitionistic Propositional Logic*

**Abstract:**

We explore two ways of formalizing Kreisel's addendum to the Brouwer-Heyting-Kolmogorov interpretation. To do this we compare Artemov's justification logic with simply typed $\lambda$ calculus. First, we provide a completeness result for Kripke-style semantics of the implicational fragment of the intuitionistic logic of proofs. Then we introduce a map from justification terms into $\lambda$ terms, which can be viewed as a method of extracting the computational content of the justification terms. Then we examine the interpretation of Kreisel's addendum in justification logic along with the image of the resulting justification terms under our map.

- - - - Wednesday, Nov 23, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York

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 *un*provability 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.

- - - - Thursday, Nov 24, 2022 - - - -

*** Thanksgiving Day ***

- - - - Friday, Nov 25, 2022 - - - -

- - - - Monday, Nov 28, 2022 - - - -

Logic and Metaphysics Workshop

Date: Monday, November 28, 4.15-6.15 (NY time), GC 7314

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

William McCarthy (Columbia).

Title: Modal pluralism and higher-order logic

Abstract: Modal pluralism is the view that there are a variety of candidate interpretations of the predicate ‘could have been the case that’ which give intuitively different answers to paradigmatic metaphysical questions (‘intuitively’ because the phrase means subtly different things on the different interpretations). It is the modal analog of set-theoretic pluralism, according to which there are a variety of candidate interpretations of ‘is a member of’. Of course, if there were a broadest kind of counterfactual possibility, then one could define every other kind as a restriction on it, as in the set-theoretic case. It would then be privileged in the way that a broadest kind of set would be, if there were one. Recently, several authors have purported to prove from higher-order logical principles that there is a broadest kind of possibility. In this talk we critically assess these arguments. We argue that they rest on an assumption which any modal pluralist should reject: namely, monism about higher-order logic. The reasons to be a modal pluralist are also reasons to be a pluralist about higher-order quantification. But from the pluralist perspective on higher-order logic, the claim that there is a broadest kind of possibility is like the Continuum Hypothesis, according to the set-theoretic pluralist. It is true on some interpretations of the relevant terminology, and false on others. Consequently, the significance of the ‘proof’ that there is a broadest kind of possibility is deflated. Time permitting, we will conclude with some upshots of higher-order pluralism for the methodology of metaphysics.

Note: This is joint work with Justin Clarke-Doane.

- - - - Tuesday, Nov 29, 2022 - - - -

- - - - Wednesday, Nov 30, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York

- - - - Thursday, Dec 1, 2022 - - - -

- - - - Friday, Dec 2, 2022 - - - -

CUNY Graduate Center, Friday, December 2, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

CUNY Graduate Center

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

Friday December 2, 2:00pm-3:30pm, Room 6417

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Wednesday seminar

## (KGRC) guests, video recordings, seminar talks Tuesday, November 22 and Thursday, November 24

## Barcelona Set Theory Seminar

ICREA Research Professor

Universitat de Barcelona

Departament de Matemàtiques i Informàtica

Gran Via de les Corts Catalanes 585

08007 Barcelona

Catalonia

Phone: +34 93 402 1609

joan.bagaria@icrea.cat

bagaria@ub.edu

## Nankai Logic Colloquium

Hello everyone,

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

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 10th Nankai Logic Colloquium -- Menachem Shlossberg

Time： 16:00pm, Nov. 18, 2022 (Beijing Time)

Zoom Number：850 1491 3444

Passcode： 596956

Link： https://us02web.zoom.us/j/85014913444?pwd=MnhUTW13MFF3S05raGEzaCs1SXhUQT09

_____________________________________________________________________

Best wishes,

Ming Xiao

## Cross-Alps Logic Seminar (speaker: Annalisa Conversano)

**Annalisa Conversano**(University of Genoa)

will give a talk on

*Tools of o-minimality in the study of groups*Please refer to the usual webpage of our LogicGroup for more details and the abstract of the talk.

The seminar will be held remotely through Webex. Please write to vincenzo.dimonte [at] uniud [dot] it for the link to the event.

The Cross-Alps Logic Seminar is co-organized by the logic groups of Genoa, Lausanne, Turin and Udine as part of our collaboration in the project PRIN 2017 'Mathematical logic: models, sets, computability'.

## This Week in Logic at CUNY

- - - - Monday, Nov 14, 2022 - - - -

Dynamics of the Knaster continuum homeomorphism group

Date: Monday, November 14, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: A new approach to Aristotle’s definitions of truth and falsehood in Metaphysics Γ.7

Abstract: At Metaphysics Γ.7, 1011b26–7, Aristotle defines truth and falsehood as follows: to assert of what is that it is or of what is not that it is not, is true; to assert of what is that it is not or of what is not that it is, is false. In their attempts to interpret the definitions, scholars usually distinguish between the veridical, 1-place, and 2-place uses of ‘to be’. The dominant view holds that all occurrences of ‘is’ in the definientia are interpreted veridically (Kahn 1966, Kirwan 1993, Crivelli 2004, Kimhi 2018, Szaif 2018). So the first truth condition is interpreted as follows: to assert of what is the case that it is the case, is true. I argue against this and side with those who favor a comprehensive—i.e. a jointly 1- and 2-place—interpretation (Matthen 1983, Wheeler 2011), according to which the first truth condition says: to assert of what is (F, exists) that it is (F, exists), is true. It is an open question how this interpretation makes Aristotle’s definitions sufficiently general so as to accommodate all propositional truth-value bearers. I first show that all Aristotelian propositions are reducible to propositions involving a 1- or 2-place ‘is’ and that formal properties, such as quantity and modality, merely modify the ‘is’, thus lending support to the comprehensive interpretation.

- - - - Tuesday, Nov 15, 2022 - - - -

Tuesday, November 15, 7:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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 -like recursively saturated models of PA.

- - - - Wednesday, Nov 16, 2022 - - - -

- - - - Thursday, Nov 17, 2022 - - - -

- - - - Friday, Nov 18, 2022 - - - -

CUNY Graduate Center, Friday, November 18, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

Brent Cody Virginia Commonwealth University

**Sparse analytic systems**

Given a set , an -predictor is a function that takes as inputs functions of the form , where , and outputs a guess for what 'should be.' An -predictor is good if for all total functions the set of for which the guess is not equal to has measure zero. Hardin and Taylor proved that every set has a good -predictor and they raised various questions asking about the extent to which the prediction made by a good predictor might be invariant after precomposing with various well-behaved functions - this leads to the notion of 'anonymity' of good predictors under various classes of functions. Bajpai and Velleman answered several of Hardin and Taylor's questions and asked: Does there exist, for every set , a good -predictor that is anonymous with respect to the strictly increasing analytic homeomorphisms of ? We provide a consistently negative answer to this question by strengthening a result of Erdős, which states that the Continuum Hypothesis is equivalent to the existence of an uncountable family of (real or complex) analytic functions, such that is countable for every . We strengthen Erdős' result by proving that CH is equivalent to the existence of what we call *sparse analytic systems* of functions. This is joint work with Sean Cox and Kayla Lee.

CUNY Graduate Center

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

Friday November 18, 2:00pm-3:30pm, Room 6417

Dima Sinapova Rutgers University

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

It is well known that if an inaccessible cardinal is singularized to countable cofinality while preserving cardinals, then 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 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 implies weak square.

- - - - Monday, Nov 21, 2022 - - - -

Logic and Metaphysics Workshop

Date: Monday, November 21, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

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.

- - - - Tuesday, Nov 22, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 22, 7:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

**Joel David Hamkins**, University of Notre Dame**Pointwise definable and Leibnizian extensions of models of arithmetic and set theory**

I shall introduce a flexible new method showing that every countable model of PA admits a pointwise definable end-extension, one in which every individual is definable without parameters. And similarly for models of set theory, in which one may also achieve the Barwise extension result—every countable model of ZF admits a pointwise definable end-extension to a model of ZFC+V=L, or indeed any theory arising in a suitable inner model. A generalization of the method shows that every model of arithmetic of size at most continuum admits a Leibnizian extension, and similarly in set theory.

- - - - Wednesday, Nov 23, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

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 *un*provability 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.

- - - - Thursday, Nov 24, 2022 - - - -

- - - - Friday, Nov 25, 2022 - - - -

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Wednesday seminar

## Upcoming Core Model Seminar

## Barcelona Set Theory Seminar

ICREA Research Professor

Universitat de Barcelona

Departament de Matemàtiques i Informàtica

Gran Via de les Corts Catalanes 585

08007 Barcelona

Catalonia

Phone: +34 93 402 1609

joan.bagaria@icrea.cat

bagaria@ub.edu

## Talk by Cristian Calude at CQT on 16 Nov 2022 16:00 hrs

## (KGRC) seminar talks Tuesday, November 15 and Thursday, November 17

## Nankai Logic Colloquium

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

## This Week in Logic at CUNY

Invariant random subgroups and characters

Date: Monday, November 7, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

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.

- - - - Tuesday, Nov 8, 2022 - - - -

Computational Logic Seminar

Fall Semester 2022, Tuesday, November 8, Time 2:00 - 4:00 PM, Room 3310-B

Speaker: Eoin Moore, Graduate Center CUNY

Title: Soundness and completeness results for LEA and probability semantics

Abstract: The goal of the logic of evidence aggregation (LEA) was to describe probabilistic evidence aggregation in the setting of formal logic. However, as noted in that paper, LEA is not complete with respect to probability semantics. This leaves open the tasks to find sound and complete semantics for LEA and a proper axiomatization for probability semantics. We do both. We define a class of basic models called deductive basic models, and show LEA is sound and complete with respect to this class. On the other side, we define an axiomatic system LEA+ extending LEA and show it is sound and complete with respect to probability semantics. Close connections to Propositional Lax Logic are also demonstrated.

Models of Peano Arithmetic (MOPA)

Tuesday, November 8, 7:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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 -like recursively saturated models of PA.

- - - - Wednesday, Nov 9, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Andrei Rodin, University of Lorraine (Nancy, France).**

Date and Time: ** Wednesday November 9, 2022, 7:00 - 8:30 PM.**

Title:** Kolmogorov's Calculus of Problems and Homotopy Type theory.**

Abstract: A. N. Kolmogorov in 1932 proposed an original version of mathematical intuitionism where the concept of problem plays a central role, and which differs in its content from the versions of intuitionism developed by A. Heyting and other followers of L. Brouwer. The popular BHK-semantics of Intuitionistic logic follows Heyting's line and conceals the original features of Kolmogorov's logical ideas. Homotopy Type theory (HoTT) implies a formal distinction between sentences and higher-order constructions and thus provides a mathematical argument in favour of Kolmogorov's approach and against Heyting's approach. At the same time HoTT does not support the constructive notion of negation applicable to general problems, which is informally discussed by Kolmogorov in the same context. Formalisation of Kolmogorov-style constructive negation remains an interesting open problem.

- - - - Thursday, Nov 10, 2022 - - - -

- - - - Friday, Nov 11, 2022 - - - -

Friday, November 11, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**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).

- - - - Monday, Nov 14, 2022 - - - -

Dynamics of the Knaster continuum homeomorphism group

Date: Monday, November 14, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: A new approach to Aristotle’s definitions of truth and falsehood in Metaphysics Γ.7

Abstract: At Metaphysics Γ.7, 1011b26–7, Aristotle defines truth and falsehood as follows: to assert of what is that it is or of what is not that it is not, is true; to assert of what is that it is not or of what is not that it is, is false. In their attempts to interpret the definitions, scholars usually distinguish between the veridical, 1-place, and 2-place uses of ‘to be’. The dominant view holds that all occurrences of ‘is’ in the definientia are interpreted veridically (Kahn 1966, Kirwan 1993, Crivelli 2004, Kimhi 2018, Szaif 2018). So the first truth condition is interpreted as follows: to assert of what is the case that it is the case, is true. I argue against this and side with those who favor a comprehensive—i.e. a jointly 1- and 2-place—interpretation (Matthen 1983, Wheeler 2011), according to which the first truth condition says: to assert of what is (F, exists) that it is (F, exists), is true. It is an open question how this interpretation makes Aristotle’s definitions sufficiently general so as to accommodate all propositional truth-value bearers. I first show that all Aristotelian propositions are reducible to propositions involving a 1- or 2-place ‘is’ and that formal properties, such as quantity and modality, merely modify the ‘is’, thus lending support to the comprehensive interpretation.

- - - - Tuesday, Nov 15, 2022 - - - -

Tuesday, November 15, 7:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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 -like recursively saturated models of PA.

- - - - Wednesday, Nov 16, 2022 - - - -

- - - - Thursday, Nov 17, 2022 - - - -

- - - - Friday, Nov 18, 2022 - - - -

CUNY Graduate Center, Friday, November 18, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

Brent Cody Virginia Commonwealth University

CUNY Graduate Center

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

Friday November 18, 2:00pm-3:30pm, Room 6417

Dima Sinapova Rutgers University

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Wednesday seminar

## (KGRC) sminar talks Tuesday, November 8 and Thursday, November 10

## Next Core Model Seminar

## Logic Seminar Wed 9 Nov 2022 17:00 hrs at NUS by Benjamin T Castle

## Logic Seminar Wed 9 Nov 2022 17:00 hrs at NUS by Benjamin T Castle

## Logic Seminar Wed 9 Nov 2022 17:00 hrs at NUS by Benjamin T Castle

## UPDATE - This Week in Logic at CUNY

Invariant random subgroups and characters

Date: Monday, Oct 31, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: The semantics of special quantification: Higher-order metaphysics and nominalization approaches

Abstract: Prior’s problem consists in the impossibility of replacing clausal complements of most attitude verbs by ‘ordinary’ NPs; only ‘special quantifiers’ that is, quantifiers like something permit a replacement, preserving grammaticality or the same reading of the verb;

(1) a. John claims that he won.

b. ??? John claims a proposition / some thing.

c. John claims something.

In my 2013 book Abstract Objects and the Semantics of Natural Language, I have shown how this generalizes to nonreferential complements of various other intensional predicates and argued for a Nominalization Theory of special quantifiers. In this talk, I will review and extend the range of linguistic generalizations that motivate the Nominalization Theory and show that they pose serious problems for a simple higher-order semantics of special quantifiers. I will outline a new version of the Nominalization Theory for special quantifiers with attitude verbs and address the question whether there can be a unified semantics of special quantifiers for the various contexts in which they display a nominalizing force.

- - - - Tuesday, Nov 1, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 1, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

**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.

- - - - Wednesday, Nov 2, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Astra Kolomatskaia, Stony Brook.**

Date and Time: ** Wednesday November 2, 2022, 7:00 - 8:30 PM. IN PERSON TALK.**

Title:** The Objective Metatheory of Simply Typed Lambda Calculus.**

Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds by constructing the normal form of a term by structural recursion on the term's syntactic representation, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Evaluation. Since the accidental discovery of the first such algorithm, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch, Hofmann, and Streicher published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. In this talk we will lay out the details of Altenkirch's seminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.

- - - - Thursday, Nov 3, 2022 - - - -

- - - - Friday, Nov 4, 2022 - - - -

CUNY Graduate Center, Friday, November 4, 12:15pm NY time, room 6495

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**Corey Switzer**, University of Vienna**The Special Tree Number**

A tree of height 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 so that if then and 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, , the least size of a tree of height which is neither non-special nor has a cofinal branch. Classical facts imply that can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that can be larger than , , and both the left hand side and bottom row of the Cichon diagram. Thus 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.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday November 4, 2:00pm-3:30pm, Room 6417

**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.

- - - - Monday, Nov 7, 2022 - - - -

Invariant random subgroups and characters

Date: Monday, November 7, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

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.

- - - - Tuesday, Nov 8, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 8, 7:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

- - - - Wednesday, Nov 9, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Andrei Rodin, University of Lorraine (Nancy, France).**

Date and Time: ** Wednesday November 9, 2022, 7:00 - 8:30 PM.**

Title:** Kolmogorov's Calculus of Problems and Homotopy Type theory.**

Abstract: A. N. Kolmogorov in 1932 proposed an original version of mathematical intuitionism where the concept of problem plays a central role, and which differs in its content from the versions of intuitionism developed by A. Heyting and other followers of L. Brouwer. The popular BHK-semantics of Intuitionistic logic follows Heyting's line and conceals the original features of Kolmogorov's logical ideas. Homotopy Type theory (HoTT) implies a formal distinction between sentences and higher-order constructions and thus provides a mathematical argument in favour of Kolmogorov's approach and against Heyting's approach. At the same time HoTT does not support the constructive notion of negation applicable to general problems, which is informally discussed by Kolmogorov in the same context. Formalisation of Kolmogorov-style constructive negation remains an interesting open problem.

- - - - Thursday, Nov 10, 2022 - - - -

- - - - Friday, Nov 11, 2022 - - - -

Friday, November 11, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Cross-Alps Logic Seminar (speaker: Jacopo Emmenegger)

**Jacopo Emmenegger**(University of Genoa)

will give a talk on

*Quotients and equality, (co)algebraically*Please refer to the usual webpage of our LogicGroup for more details and the abstract of the talk.

The seminar will be held remotely through Webex. Please write to vincenzo.dimonte [at] uniud [dot] it for the link to the event.

The Cross-Alps Logic Seminar is co-organized by the logic groups of Genoa, Lausanne, Turin and Udine as part of our collaboration in the project PRIN 2017 'Mathematical logic: models, sets, computability'.

## This Week in Logic at CUNY

Invariant random subgroups and characters

Date: Monday, Oct 31, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: The semantics of special quantification: Higher-order metaphysics and nominalization approaches

Abstract: Prior’s problem consists in the impossibility of replacing clausal complements of most attitude verbs by ‘ordinary’ NPs; only ‘special quantifiers’ that is, quantifiers like something permit a replacement, preserving grammaticality or the same reading of the verb;

(1) a. John claims that he won.

b. ??? John claims a proposition / some thing.

c. John claims something.

In my 2013 book Abstract Objects and the Semantics of Natural Language, I have shown how this generalizes to nonreferential complements of various other intensional predicates and argued for a Nominalization Theory of special quantifiers. In this talk, I will review and extend the range of linguistic generalizations that motivate the Nominalization Theory and show that they pose serious problems for a simple higher-order semantics of special quantifiers. I will outline a new version of the Nominalization Theory for special quantifiers with attitude verbs and address the question whether there can be a unified semantics of special quantifiers for the various contexts in which they display a nominalizing force.

- - - - Tuesday, Nov 1, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 1, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

**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.

- - - - Wednesday, Nov 2, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Astra Kolomatskaia, Stony Brook.**

Date and Time: ** Wednesday November 2, 2022, 7:00 - 8:30 PM. IN PERSON TALK.**

Title:** The Objective Metatheory of Simply Typed Lambda Calculus.**

Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds by constructing the normal form of a term by structural recursion on the term's syntactic representation, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Evaluation. Since the accidental discovery of the first such algorithm, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch, Hofmann, and Streicher published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. In this talk we will lay out the details of Altenkirch's seminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.

- - - - Thursday, Nov 3, 2022 - - - -

- - - - Friday, Nov 4, 2022 - - - -

CUNY Graduate Center, Friday, November 4, 12:15pm NY time, room 6495

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**Corey Switzer**, University of Vienna**The Special Tree Number**

A tree of height 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 so that if then and 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, , the least size of a tree of height which is neither non-special nor has a cofinal branch. Classical facts imply that can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that can be larger than , , and both the left hand side and bottom row of the Cichon diagram. Thus 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.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday November 4, 2:00pm-3:30pm, Room 6417

**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.

- - - - Monday, Nov 7, 2022 - - - -

Invariant random subgroups and characters

Date: Monday, November 7, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

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.

- - - - Tuesday, Nov 8, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 8, 7:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

- - - - Wednesday, Nov 9, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Andrei Rodin, University of Lorraine (Nancy, France).**

Date and Time: ** Wednesday November 9, 2022, 7:00 - 8:30 PM.**

Title:** Kolmogorov's Calculus of Problems and Homotopy Type theory.**

Abstract: A. N. Kolmogorov in 1932 proposed an original version of mathematical intuitionism where the concept of problem plays a central role, and which differs in its content from the versions of intuitionism developed by A. Heyting and other followers of L. Brouwer. The popular BHK-semantics of Intuitionistic logic follows Heyting's line and conceals the original features of Kolmogorov's logical ideas. Homotopy Type theory (HoTT) implies a formal distinction between sentences and higher-order constructions and thus provides a mathematical argument in favour of Kolmogorov's approach and against Heyting's approach. At the same time HoTT does not support the constructive notion of negation applicable to general problems, which is informally discussed by Kolmogorov in the same context. Formalisation of Kolmogorov-style constructive negation remains an interesting open problem.

- - - - Thursday, Nov 10, 2022 - - - -

- - - - Friday, Nov 11, 2022 - - - -

Friday, November 11, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Barcelona Set Theory Seminar

ICREA Research Professor

Universitat de Barcelona

Departament de Matemàtiques i Informàtica

Gran Via de les Corts Catalanes 585

08007 Barcelona

Catalonia

Phone: +34 93 402 1609

joan.bagaria@icrea.cat

bagaria@ub.edu

## Core Model Seminar on Tuesday

## (KGRC) two(!) seminar talks on Thursday, November 3

## Wednesday seminar

## Logic Seminar Wed 2 Nov 2022 17:00 hrs at NUS by Wu Guohua

## Logic Seminar 26 October 2022 17:00 hrs at NUS by Sun Mengzhou

## UPDATE - This Week in Logic at CUNY

The Special Tree Number

Logic and Metaphysics Workshop

Date: Monday, Oct 24, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Tomorrow, Monday, October 24th, 4.15-6.15 (NY time)

Speaker: Rohit Parikh (CUNY)

Speaker Medium: In-person at the Graduate Center, Room 7314 (you may also attend virtually).

Title: A measure of group coherence

- - - - Tuesday, Oct 25, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, October 25, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

**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.

Tuesday, October 25, Time 2:00 - 4:00 PM, Room 3310-B,

For a zoom link contact SArtemov@gmail.com

Title: How to Prove, and Not to Prove, Consistency

Abstract: The consistency of a formal theory is a sequential property

**C**= {C_0, C_1, ... C_n, ...}, where each C_n states that the n-th derivation does not contain a contradiction. For proving

**C**in a theory

**T**, Hilbert suggested (i) finding a procedure that given n builds a

**T**-proof of C_n and (ii) proving in

**T**that this procedure always works.

However, for Peano Arithmetic PA, the traditional way here has been to compress **C** into a single arithmetical formula Consis(PA) and apply the Second Gödel Incompleteness theorem, stating the unprovability of Consis(PA) in PA, to claim the unprovability of **C** in PA. This chain of reasoning is fundamentally flawed: one can only conclude that (a compressed form of) consistency is not provable in a FINITE fragment of PA whereas PA is known to be (much) stronger than any of its finite fragments.

Following the original Hilbert's approach, we were able to show that the consistency property of PA is indeed provable in PA. These findings dismantle a foundational “impossibility paradigm”: *there exists no consistency proof of a system that can be formalized in the system itself.* (Encyclopaedia Britannica, Article "Metalogic," 2000).

- - - - Wednesday, Oct 26, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Ross Street, Macquarie University.**

Date and Time: ** Wednesday October 26, 2022, 7:00 - 8:30 PM.**

Title:** The core groupoid can suffice.**

Abstract: Let V be the monoidal category of modules over a commutative ring R. I am interested in categories A for which there is a groupoid G such that the functor categories [A,V] and [G,V] are equivalent. In particular, G could be the core groupoid of A; that is, the subcategory with the same objects and with only the invertible morphisms. Every category A can be regarded as a V-category (that is, an R-linear category), denoted RA, with the same objects and with hom R-module RA(a,b) free on the homset A(a,b). Indeed, RA is the free V-category on A so that the V-functor category [RA,V] is the ordinary functor category [A,V] with the pointwise R-linear structure. In these terms, we are interested in when RA and RG are Morita equivalent V-categories. In my joint work with Steve Lack on Dold-Kan-type equivalences, we had many examples of this phenomenon. However, the example of Nick Kuhn, where A is the category of finite vector spaces over a fixed finite field F with all F-linear functions and G is the general linear groupoid over F, does not fit our theory. Yet the ``kernel'' of the equivalence is of the same type. The present work shows that the category theory behind the Kuhn result also covers our Dold-Kan-type setting. I plan to start with a baby example which highlights the ideas.

I am grateful to Nick Kuhn and Ben Steinberg for their patient email correspondence with me on this topic.

- - - - Thursday, Oct 27, 2022 - - - -

- - - - Friday, Oct 28, 2022 - - - -

CUNY Graduate Center, Friday, October 28, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**Forcing ' is -dense' from Large Cardinals - A Journey guided by the Stars: Part II**

An ideal on is -dense if has a dense subset of size . We prove, assuming large cardinals, that there is a semiproper forcing so thatThis answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of in Woodin's axiom by results in an axiom which implies .

We proceed in three steps: First we define and motivate a new forcing axiom and then modify the Asperó-Schindler proof of to show . Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing . This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday October 28, 2:00pm-3:30pm, Room 6417

**Ideal Independence, Filters and Maximal Sets of Reals**

A family is called ideal independent if given any finite, distinct , the set is infinite. In other words, the ideal generated by does not contain for any . The least size of a maximal (with respect to inclusion) ideal independent family is denoted 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 . 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.

- - - - Monday, Oct 31, 2022 - - - -

Invariant random subgroups and characters

Date: Monday, Oct 31, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: The semantics of special quantification: Higher-order metaphysics and nominalization approaches

Abstract: Prior’s problem consists in the impossibility of replacing clausal complements of most attitude verbs by ‘ordinary’ NPs; only ‘special quantifiers’ that is, quantifiers like something permit a replacement, preserving grammaticality or the same reading of the verb;

(1) a. John claims that he won.

b. ??? John claims a proposition / some thing.

c. John claims something.

In my 2013 book Abstract Objects and the Semantics of Natural Language, I have shown how this generalizes to nonreferential complements of various other intensional predicates and argued for a Nominalization Theory of special quantifiers. In this talk, I will review and extend the range of linguistic generalizations that motivate the Nominalization Theory and show that they pose serious problems for a simple higher-order semantics of special quantifiers. I will outline a new version of the Nominalization Theory for special quantifiers with attitude verbs and address the question whether there can be a unified semantics of special quantifiers for the various contexts in which they display a nominalizing force.

- - - - Tuesday, Nov 1, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 1, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

- - - - Wednesday, Nov 2, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Astra Kolomatskaia, Stony Brook.**

Date and Time: ** Wednesday November 2, 2022, 7:00 - 8:30 PM. IN PERSON TALK.**

Title:** The Objective Metatheory of Simply Typed Lambda Calculus.**

Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds by constructing the normal form of a term by structural recursion on the term's syntactic representation, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Evaluation. Since the accidental discovery of the first such algorithm, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch, Hofmann, and Streicher published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. In this talk we will lay out the details of Altenkirch's seminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.

- - - - Thursday, Nov 3, 2022 - - - -

- - - - Friday, Nov 4, 2022 - - - -

CUNY Graduate Center, Friday, November 4, 12:15pm NY time, room 6495

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**Corey Switzer**, University of Vienna**The Special Tree Number**

A tree of height 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 so that if then and 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, , the least size of a tree of height which is neither non-special nor has a cofinal branch. Classical facts imply that can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that can be larger than , , and both the left hand side and bottom row of the Cichon diagram. Thus 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.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday November 4, 2:00pm-3:30pm, Room 6417

**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.

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## This Week in Logic at CUNY

The Special Tree Number

Logic and Metaphysics Workshop

Date: Monday, Oct 24, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Tomorrow, Monday, October 24th, 4.15-6.15 (NY time)

Speaker: Rohit Parikh (CUNY)

Speaker Medium: In-person at the Graduate Center, Room 7314 (you may also attend virtually).

Title: A measure of group coherence

- - - - Tuesday, Oct 25, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, October 25, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

Tuesday, October 25, Time 2:00 - 4:00 PM, Room 3310-B,

For a zoom link contact SArtemov@gmail.com

Title: How to Prove, and Not to Prove, Consistency

Abstract: The consistency of a formal theory is a sequential property

**C**= {C_0, C_1, ... C_n, ...}, where each C_n states that the n-th derivation does not contain a contradiction. For proving

**C**in a theory

**T**, Hilbert suggested (i) finding a procedure that given n builds a

**T**-proof of C_n and (ii) proving in

**T**that this procedure always works.

However, for Peano Arithmetic PA, the traditional way here has been to compress **C** into a single arithmetical formula Consis(PA) and apply the Second Gödel Incompleteness theorem, stating the unprovability of Consis(PA) in PA, to claim the unprovability of **C** in PA. This chain of reasoning is fundamentally flawed: one can only conclude that (a compressed form of) consistency is not provable in a FINITE fragment of PA whereas PA is known to be (much) stronger than any of its finite fragments.

Following the original Hilbert's approach, we were able to show that the consistency property of PA is indeed provable in PA. These findings dismantle a foundational “impossibility paradigm”: *there exists no consistency proof of a system that can be formalized in the system itself.* (Encyclopaedia Britannica, Article "Metalogic," 2000).

- - - - Wednesday, Oct 26, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Ross Street, Macquarie University.**

Date and Time: ** Wednesday October 26, 2022, 7:00 - 8:30 PM.**

Title:** The core groupoid can suffice.**

Abstract: Let V be the monoidal category of modules over a commutative ring R. I am interested in categories A for which there is a groupoid G such that the functor categories [A,V] and [G,V] are equivalent. In particular, G could be the core groupoid of A; that is, the subcategory with the same objects and with only the invertible morphisms. Every category A can be regarded as a V-category (that is, an R-linear category), denoted RA, with the same objects and with hom R-module RA(a,b) free on the homset A(a,b). Indeed, RA is the free V-category on A so that the V-functor category [RA,V] is the ordinary functor category [A,V] with the pointwise R-linear structure. In these terms, we are interested in when RA and RG are Morita equivalent V-categories. In my joint work with Steve Lack on Dold-Kan-type equivalences, we had many examples of this phenomenon. However, the example of Nick Kuhn, where A is the category of finite vector spaces over a fixed finite field F with all F-linear functions and G is the general linear groupoid over F, does not fit our theory. Yet the ``kernel'' of the equivalence is of the same type. The present work shows that the category theory behind the Kuhn result also covers our Dold-Kan-type setting. I plan to start with a baby example which highlights the ideas.

I am grateful to Nick Kuhn and Ben Steinberg for their patient email correspondence with me on this topic.

- - - - Thursday, Oct 27, 2022 - - - -

- - - - Friday, Oct 28, 2022 - - - -

CUNY Graduate Center, Friday, October 28, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**Forcing ' is -dense' from Large Cardinals - A Journey guided by the Stars: Part II**

An ideal on is -dense if has a dense subset of size . We prove, assuming large cardinals, that there is a semiproper forcing so thatThis answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of in Woodin's axiom by results in an axiom which implies .

We proceed in three steps: First we define and motivate a new forcing axiom and then modify the Asperó-Schindler proof of to show . Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing . This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday October 28, 2:00pm-3:30pm, Room 6417

**Ideal Independence, Filters and Maximal Sets of Reals**

A family is called ideal independent if given any finite, distinct , the set is infinite. In other words, the ideal generated by does not contain for any . The least size of a maximal (with respect to inclusion) ideal independent family is denoted 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 . 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.

- - - - Monday, Oct 31, 2022 - - - -

Invariant random subgroups and characters

Date: Monday, Oct 31, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: The semantics of special quantification: Higher-order metaphysics and nominalization approaches

Abstract: Prior’s problem consists in the impossibility of replacing clausal complements of most attitude verbs by ‘ordinary’ NPs; only ‘special quantifiers’ that is, quantifiers like something permit a replacement, preserving grammaticality or the same reading of the verb;

(1) a. John claims that he won.

b. ??? John claims a proposition / some thing.

c. John claims something.

In my 2013 book Abstract Objects and the Semantics of Natural Language, I have shown how this generalizes to nonreferential complements of various other intensional predicates and argued for a Nominalization Theory of special quantifiers. In this talk, I will review and extend the range of linguistic generalizations that motivate the Nominalization Theory and show that they pose serious problems for a simple higher-order semantics of special quantifiers. I will outline a new version of the Nominalization Theory for special quantifiers with attitude verbs and address the question whether there can be a unified semantics of special quantifiers for the various contexts in which they display a nominalizing force.

- - - - Tuesday, Nov 1, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, November 1, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

- - - - Wednesday, Nov 2, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

Speaker: ** Astra Kolomatskaia, Stony Brook.**

Date and Time: ** Wednesday November 2, 2022, 7:00 - 8:30 PM. IN PERSON TALK.**

Title:** The Objective Metatheory of Simply Typed Lambda Calculus.**

- - - - Thursday, Nov 3, 2022 - - - -

- - - - Friday, Nov 4, 2022 - - - -

CUNY Graduate Center, Friday, November 4, 12:15pm NY time, room 6495

Hybrid: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**Corey Switzer**, University of Vienna**The Special Tree Number**

*special* if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function so that if then and 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, , the least size of a tree of height which is neither non-special nor has a cofinal branch. Classical facts imply that can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that can be larger than , , and both the left hand side and bottom row of the Cichon diagram. Thus 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.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday October 28, 2:00pm-3:30pm, Room 6417

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

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Nankai Logic Colloquium

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

___________________________________________________________________________________________________________________________________________________

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

## Wednesday seminar

## Barcelona Set Theory Seminar

ICREA Research Professor

Universitat de Barcelona

Departament de Matemàtiques i Informàtica

Gran Via de les Corts Catalanes 585

08007 Barcelona

Catalonia

Phone: +34 93 402 1609

joan.bagaria@icrea.cat

bagaria@ub.edu

## Upcoming Core Model Seminar

## (KGRC) Set Theory Seminar talk Tuesday, October 25

## Cross-Alps Logic Seminar (speaker: Christian Rosendal)

**Christian Rosendal**(University of Maryland)

will give a talk on

*Amenability, optimal transport and complementation in Banach modules*Please refer to the usual webpage of our LogicGroup for more details and the abstract of the talk.

The seminar will be held remotely through Webex. Please write to vincenzo.dimonte [at] uniud [dot] it for the link to the event.

The Cross-Alps Logic Seminar is co-organized by the logic groups of Genoa, Lausanne, Turin and Udine as part of our collaboration in the project PRIN 2017 'Mathematical logic: models, sets, computability'.

## This Week in Logic at CUNY

- - - - Monday, Oct 17, 2022 - - - -

Rutgers Logic Seminar

Abstract: Over the past few years, many exciting connections have been found between descriptive combinatorics, where one studies classical combinatorial problems under definability constraints such as measurability or Baire-measurability, and other combinatorial settings. These include the LOCAL model of Linial used in the theory of distributed computing, and the theory of random processes. The subject of this talk is the more recent addition of computable combinatorics, where the definability constraints are, e.g. computability or recursive enumerability, to this picture. We describe several parallels between this computable setting and the Baire-measurable setting, and show how a geometric structure known as “toast” gives a systematic way of explaining them. As an application, we adapt a proof of Kierstead from the computable setting to get new upper bounds on Baire-measurable edge chromatic numbers. We also show that, for acyclic graphs, the “local problems” which can be solved Baire-measurably are exactly those which can be solved computably on so called “highly computable” graphs.

Logic and Metaphysics Workshop

Date: Monday, Oct 17, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Note: This is joint work with John Lane Bell.

- - - - Tuesday, Oct 18, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, October 18, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

Say that a theory is tight if any two distinct extensions of cannot be bi-interpretable. Vaguely speaking, tightness expresses a sort of maximality to the expressiveness of . Visser showed that is tight and building on this work, Enayat showed that , 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 . The answer to this question is positive. If you restrict the comprehension axiom of to only arithmetical formulae, or if you restrict it to formulae, the resulting theory is not tight. As a specific instance, we show that if is either the minimum omega-model of or the minimum beta-model of - for some , then is bi-interpretable with a carefully chosen extension 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.

- - - - Wednesday, Oct 19, 2022 - - - -

Department of Computer Science

Department of Mathematics

The Graduate Center of The City University of New York, Room 6417

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.

- - - - Thursday, Oct 20, 2022 - - - -

- - - - Friday, Oct 21, 2022 - - - -

CUNY Graduate Center, Friday, October 21, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

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

An ideal on is -dense if has a dense subset of size . We prove, assuming large cardinals, that there is a semiproper forcing so thatThis answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of in Woodin's axiom by results in an axiom which implies .

We proceed in three steps: First we define and motivate a new forcing axiom and then modify the Asperó-Schindler proof of to show . Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing . This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday October 21, 2:00pm-3:30pm, Room 6417

Philipp Rothmaler, CUNY

**Generalized Bass modules**

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.

- - - - Monday, Oct 24, 2022 - - - -

- - - - Tuesday, Oct 25, 2022 - - - -

- - - - Wednesday, Oct 26, 2022 - - - -

- - - - Thursday, Oct 27, 2022 - - - -

- - - - Friday, Oct 28, 2022 - - - -

CUNY Graduate Center, Friday, October 28, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

**Forcing ' is -dense' from Large Cardinals - A Journey guided by the Stars: Part II**

We proceed in three steps: First we define and motivate a new forcing axiom and then modify the Asperó-Schindler proof of to show . Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing . This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday October 28, 2:00pm-3:30pm, Room 6417

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

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## (KGRC) Set Theory Seminar talk Tuesday, October 18

## Wednesday seminar

## Logic Seminar 19 Oct 2022 17:00 hrs at NUS by Abdul Basit

## Logic Seminar 12 October 2022 17:00 hrs at NUS by Frank Stephan

## Barcelona Set Theory Seminar (change of date)

ICREA Research Professor

Universitat de Barcelona

Departament de Matemàtiques i Informàtica

Gran Via de les Corts Catalanes 585

08007 Barcelona

Catalonia

Phone: +34 93 402 1609

joan.bagaria@icrea.cat

bagaria@ub.edu

## Barcelona Set Theory Seminar

ICREA Research Professor

Universitat de Barcelona

Departament de Matemàtiques i Informàtica

Gran Via de les Corts Catalanes 585

08007 Barcelona

Catalonia

Phone: +34 93 402 1609

joan.bagaria@icrea.cat

bagaria@ub.edu

## This Week in Logic at CUNY

- - - - Monday, Oct 10, 2022 - - - -

*** Graduate Center Closed Today ***

- - - - Tuesday, Oct 11, 2022 - - - -

Tuesday, October 11, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

**Fedor Pakhomov**, Ghent University**How to escape Tennenbaum's theorem**

We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model. See my preprint https://arxiv.org/abs/2209.00967 for more details.

Tuesday, October 11, Time 2:00 - 4:00 PM

Room 3310-B

Speaker: Melvin Fitting, Graduate Center

Title: Applying Tableaus to Observable Models and Hypertheories

My talk will begin with a brief introduction to tableaus, for those not particularly familiar with them. Following recent advice from a talk in the seminar itself, this will begin with classical logic. Then we move to versions appropriate for modal logics. We will see that we already have, in the literature, useful machinery for investigating hypertheories and their models. The machinery has been available for a long time. The fact that it applies with no changes says something about the naturalness of hypertheories.

- - - - Wednesday, Oct 12, 2022 - - - -

- - - - Thursday, Oct 13, 2022 - - - -

- - - - Friday, Oct 14, 2022 - - - -

CUNY Graduate Center, Friday, October 14, 12:15pm NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday October 14, 2:00pm-3:30pm, Room 6417

**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.

- - - - Monday, Oct 17, 2022 - - - -

Logic and Metaphysics Workshop

Date: Monday, Oct 17, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Note: This is joint work with John Lane Bell.

- - - - Tuesday, Oct 18, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, October 18, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

Say that a theory is tight if any two distinct extensions of cannot be bi-interpretable. Vaguely speaking, tightness expresses a sort of maximality to the expressiveness of . Visser showed that is tight and building on this work, Enayat showed that , 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 . The answer to this question is positive. If you restrict the comprehension axiom of to only arithmetical formulae, or if you restrict it to formulae, the resulting theory is not tight. As a specific instance, we show that if is either the minimum omega-model of or the minimum beta-model of - for some , then is bi-interpretable with a carefully chosen extension by Cohen-forcing.

This talk is about joint work with Alfredo Roque Freire.

- - - - Wednesday, Oct 19, 2022 - - - -

- - - - Thursday, Oct 20, 2022 - - - -

- - - - Friday, Oct 21, 2022 - - - -

CUNY Graduate Center

Hybrid (email Victoria Gitman for meeting id)

Friday October 21, 2:00pm-3:30pm, Room 6417

Philipp Rothmaler, CUNY

**Generalized Bass modules**

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.

- - - - Other Logic News - - - -

- - - - Web Site - - - -

Find us on the web at: nylogic.github.io

(site designed, built & maintained by Victoria Gitman)

-------- ADMINISTRIVIA --------

To subscribe/unsubscribe to this list, please email your request to jreitz@nylogic.org.

If you have a logic-related event that you would like included in future mailings, please email jreitz@nylogic.org.

## Math Logic Seminar this Tuesday

## Wednesday seminar

## (KGRC) Set Theory Research Seminar talk Tuesday, October 11

## Core Model Seminar

## This Week in Logic at CUNY

Logic and Metaphysics Workshop

Date: Monday, Sept 19, 4.15-6.15 (NY time), GC 7315

For meeting information (including zoom link for those wishing to attend remotely), please sign up for our mailing list at https://logic.commons.gc.cuny.edu/about/

Title: The best of all possible Leibnizian completeness theorems

Abstract: Leibniz developed several arithmetical interpretations of the assertoric syllogistic in a series of papers from April 1679. In this talk, I present his most mature arithmetical semantics. I show that the assertoric syllogistic can be characterized exactly not only in the full divisibility lattice, as Leibniz implicitly suggests, but in a certain four-element sublattice thereof. This refinement is also shown to be optimal in the sense that the assertoric syllogistic is not complete with respect to any smaller sublattice using Leibniz’s truth conditions.

- - - - Tuesday, Oct 4, 2022 - - - -

Models of Peano Arithmetic (MOPA)

Tuesday, October 4, 1:00pm

Virtual (email Victoria Gitman vgitman@nylogic.org for meeting id)

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

It is well known that every countable recursively saturated model of has a full compositional truth predicate; that is, such a model is expandable to the theory . It is also well known that such a truth predicate need not be inductive, or indeed, need not satisfy even induction. Recently, Enayat and Pakhomov showed that induction for the truth predicate is equivalent to the principle of disjunctive correctness: the assertion that for any sequence of sentences , the disjunction is evaluated as true if and only if there is such that is evaluated as true. In the absence of induction, various pathologies can occur, including models of 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 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 if and only if 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.

- - - - Wednesday, Oct 5, 2022 - - - -

- - - - Thursday, Oct 6, 2022 - - - -

- - - - Friday, Oct 7, 2022 - - - -

CUNY Graduate Center, Friday, October 7, 11am NY time

Virtual: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.

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

For a class of posets, a cardinal is said to be *generically supercompact by * (or *-gen. supercompact* for short) if, for any , there are such that, for all -generic there are , with , , and .

A cardinal is *Laver-generically supercompact for * (or *-Laver-gen. supercompact* for short) if, for any , and -generic , there are -name with such that, for all -generic , there are , such that , , and , , .

-gen. superhuge, and -Laver-gen. superhuge cardinals are defined if the condition is replaced with .

Perhaps it is not apparent at first sight in the formulation the definitions above but these notions of generic large cardinals are first-order definable (S.F, and H. Sakai [1]).

While the generic supercompactness does not determine the size of the cardinal. Laver-generic supercompactness determines the size of the cardinal __and__ that of the continuum in most of the natural settings of (see S.F., A.Ottenbreit Maschio Rodrigues, and H. Sakai [0] for a proof):

(A) If is -Laver-gen. supercompact for a class of posets such that (1) all are -preserving, (2) all do not add reals, and (3) there is a which collapses , __then__ and CH holds.

(B) If is -Laver-gen. supercompact for a class